Two notes were sent to a local television station days after the 84-year-old went missing from her home outside Tucson in the US state of Arizona on 1 February.
The Pima County Sheriff's Department said it believes "these distinctive linguistic characteristics may be recognisable to someone who knows or has interacted with the notes' writer".
The first note demanded $4m (£3m) in bitcoin for Ms Guthrie's release. Here is the text in full:
"Hello Savannah,
"We have your mother Nancy. She is safe but scared. She will be held for ransom and once payment is received she will be released unharmed.
"We will be holding her for a maximum of 7 days. You will need to send bitcoin in the amount of $4 million USD before 5PM on Thursday the 5th.
"If payment is not received, the ransom will be increased to 6 million USD which will need to be paid by 5PM on Monday the 9th.
"Once payment is received to the bitcoin address below she will be released within 12 hours of deposit to a safe drop off location back in Tucson.
"If payment is not recieved (sic) by the last deadline on Monday the 9th at 5PM, she will be killed. Your mother is aware of this and her life is in your hands. It is in the best interest of everyone to have this completed as soon as possible.
"You will not be able to contact me from here on out, there will be no negotiation. Do not play games, law enforcement will not be able to help you.
"She had a white smart watch on the floor of the foot of her bed and the white flood light in backyard was destroyed."
The second note claimed Ms Guthrie had "perished shortly after she was taken". Here is the text in full:
"Guthrie Family,
"We did not fully grasp the seriousness of her physical condition. We never intended to hurt her, that was not our intention.
"She perished shortly after she was taken. We believe it was heart related. She is buried in nature now.
"Nothing you could have done could have changed the outcome. We want your family to know this and hope you all can find peace. We are truly sorry."
The Pima County Sheriff's Department is urging anyone who may recognise the patterns in the notes to contact them, saying in a statement on Friday: "It is not too late to come forward."
It also said two videos recovered from Nancy Guthrie's doorbell camera might be from two separate days and indicate an unidentified male prepared ahead of the kidnapping.
The FBI has previously released images of a suspect in the case.
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The duet has now spent 19 non-consecutive weeks at number one in the Official Charts, breaking a record set more than 70 years ago in 1953, by Frankie Laine's I Believe.
Rein Me In is a song by Fender which featured on his third album, last year's Mercury Prize winner People Watching.
It was re-released as a duet with Dean in June 2025. A slow-burner, it hovered mainly in the top 20 for months before first topping the chart in February this year.
That ascent in itself gave the song a record - for the longest climb to the top in UK chart history.
Since its first week at number one, the song has only shifted down on a handful of occasions, thanks to releases by heavy-hitters Harry Styles, Olivia Rodrigo, Ariana Grande, and Taylor Swift, but it has always made it back to the summit.
Accepting a special record breaker award as Rein Me In held off competition from Shakira and Burna Boy's World Cup hit Dai Dai to stay in the top spot once again this week, Fender told Official Charts he was "completely lost for words".
He also thanked Dean and all their fans "for making this song the juggernaut that it is".
Rein Me In was named song of the year at the Brit Awards, winning over other shortlisted tracks including big hits such as Raye's Where Is My Husband!, Messy, by Lola Young, and Dean's own Man I Need.
Dean was the big winner at the ceremony, taking home four gongs in total. She also collected her first Grammy at the beginning of the year.
Earlier this week, her 2025 album, The Art Of Loving, was announced as one of the 12 shortlisted for this year's Mercury Prize - which Fender won last year.
The big chart hitters
Famously in the 1990s, there were two huge long-running chart hits - Bryan Adams' 1991 hit (Everything I Do) I Do It For You, from the Robin Hood: Prince Of Thieves soundtrack, and Wet Wet Wet's cover of Love Is All Around, from Four Weddings And A Funeral.
Adams' song, which topped the UK chart for 16 weeks in a row in 1991, remains the longest consecutively-reigning number one.
Earlier this month, when Rein Me In took the record for longest-running British number one from Wet Wet Wet, the band - now made up of Graeme Clark, Graeme Duffin and vocalist Kevin Simm - congratulated the duo, telling Sky News: "It's great to see amazing British artists continuing to produce songs that connect with so many people."
Other songs to have spent more than 10 weeks at the top spot in the UK, consecutively or otherwise, include One Dance by Drake, Queen's Bohemian Rhapsody, Dance Monkey by Tones And I, and two for Ed Sheeran - Shape Of You and Bad Habits.
Congratulating Fender and Dean on their success, Becca Monahan and Chris Austin, interim co-managing directors of Official Charts, said it was "an extraordinary, history-making" achievement.
"For Rein Me In to break a record that has stood for 73 years is remarkable in itself, but it feels especially significant to see two homegrown British artists write a new chapter in Official Chart history together," they said.
"Sam and Olivia are both exceptional talents, and the incredible connection Rein Me In has made with audiences speaks to the strength and ambition of British music today."
Danny Cahalane, 38, was taken to hospital after he was attacked at his home in Lipson Road, Plymouth, while his young daughter was asleep on 21 February 2025.
The father-of-two died in hospital on 3 May, where he had been treated for 10 weeks for severe burns caused by sulphuric acid in the early-hours assault.
His ex-wife, Paris Wilson, supplied information on his whereabouts to a crime boss and another organiser, Ramarnee Bakas.
Israel Augustus and Abdul-Rasheed Adedoja were convicted of his murder, with police saying Adedoja was instrumental in planning the killing and directing events that followed.
Before his death, Mr Cahalane was able to tell police he believed his drugs "boss", referred to by his nickname Frost in the trial, had ordered the fatal attack, jurors were told.
Wilson, 35, of Plymouth, Devon, wiped tears from her eyes at Winchester Crown Court as she was jailed for nine years following an 18-week trial. She was convicted of manslaughter, and for her part in an earlier kidnap attempt against Mr Cahalane in January 2025.
Adedoja, 23, of Neasden, north London, and Augustus, 27, of Tottenham, north London, were found guilty of murdering Mr Cahalane.
Both were jailed for life to serve a minimum of 31 and 30 years respectively.
A character reference submitted by Adedoja's sister was dismissed by the judge, who said it appeared to have been written with the assistance of AI.
She said submissions like this had become increasingly common, and that she had seen one before with the words "insert defendant's name" still included.
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Augustus was also sentenced for possessing a zombie-style knife in a private place and drugs offences.
Bakas, 24, of London, was also jailed for 10 years for manslaughter.
The judge, Ms Justice Norton, said that Wilson had "used her child effectively as bait" for the attempted kidnap, which she said was a "particularly unpleasant" aspect of the case.
She told Wilson: "Whilst you yourself did not deal drugs, from evidence placed before the jury and messages to your mother, you were clearly happy to associate and provide assistance to those who do.
"That assistance provided details to Frost of Danny Cahalane's whereabouts thereby allowing the attack to take place.
"You also used contact arrangements with your daughter to set up Danny Cahalane for an attempted kidnap a month earlier."
The judge told Wilson that there were "two sides to her character", praising the defendant's "hard-working" nature and efforts in bringing up her children.
But, turning to Wilson's involvement in the attack, she said: "Whatever your motivation, whether money, security or revenge, in that moment your main concern was yourself."
Mr Cahalane had worked for a drugs boss based overseas, managing the Plymouth branch of his operation, and owed him around £120,000.
He had run up the debts after one of his junior drug dealers fled without paying him, with more money lost through gambling, and he was unable to repay the Dubai-based kingpin.
Jo Martin KC, prosecuting, told the court this boss was a man called Ryan Kennedy, nicknamed Frost.
She said Mr Cahalane had split "acrimoniously" with Wilson by 2025, and she became Mr Kennedy's "go-to contact in Plymouth to try to put pressure on Danny to pay up, and then to get information about Danny's whereabouts".
Sentencing, the judge said that Mr Cahalane was attacked because "he owed a significant drug debt", adding he "suffered horrific injuries after he was doused in acid".
She said: "Danny suffered significant corrosive injuries that had he survived would have been life-changing.
"For most if not all of the weeks between the assault and his death he would have been in significant pain."
Detective Superintendent Rachael Bentley, from Devon and Cornwall Police, said: "This was a particularly barbaric attack on a man in his own home - a place he should have been safe.
"It involved numerous individuals based in Plymouth, London and abroad who performed various roles in the planning, execution and events following the attack.
"Organised crime has no place in our community and when it does creep into our force area, we will investigate and target those responsible regardless of where they stand in the chain of command."
The rocket's upper stage will unintentionally slam into the moon next week, carving out a crater and sending up a plume of dust and rubble that scientists and skygazers are eager to observe.
Space tracking expert Bill Gray predicts an impact speed of 5,400mph - seven times the speed of sound - near Einstein Crater on the moon's sunlit western limb when it hits on Wednesday.
The rocket was used to launch a pair of lunar landers more than a year ago, and while scientists are not too concerned about the debris, it highlights the growing threat as more items head into orbit.
"Things are getting crowded up there," said Mr Gray, who plans to view the aftermath from New Brunswick, Canada.
It was never SpaceX's intention to hit the moon, but experts said the crash could have been avoided if the upper stage had been nudged into orbit around the sun.
It will be the second dead rocket known to crash into the moon accidentally, after a Chinese rocket segment left a pair of craters on the lunar far side in 2022.
Luckily for astronomers, the upcoming collision will occur on the moon's near side.
The explosion will have energy equivalent to three tons of TNT, but the impact flash - lasting less than a second - will probably be too dim to see, according to experts.
Nonetheless the stream of ejected material could stretch for several miles into space and remain visible to telescopes for tens of minutes.
The eastern portions of the US and Canada, and much of South America should have the best views.
"The gravity on the moon is low and there is no wind to blow the dust away," said Benjamin Fernando of Los Alamos National Laboratory, who's encouraging observations by professionals and amateurs alike.
He added: "Part of the reason for our interest in this event is to figure out how much of a hazard debris impacts pose to future astronauts."
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Dr Fernando anticipates there will be an impact crater nearly 27m (90ft) across and 5m (16ft) deep, too small to see from Earth but visible to spacecraft.
NASA's Lunar Reconnaissance Orbiter and South Korea's Danuri lunar orbiter will gather before-and-after shots of the crash scene.
Danuri will stray within a mile or two of the rocket just two minutes before impact, according to Mr Fernando and his team.
The abandoned rocket segment - measuring some 12m (40ft) and weighing around 4,500kg - hoisted two private lunar landers on 15 January 2025.
One of them - Firefly Aerospace's Blue Ghost - became the first private spacecraft to pull off a fully-successful lunar touchdown. The other - a lander from Japan-based ispace - wrecked.
With meteoroids and other natural objects offering little if any warning before walloping the moon, scientists said there's much to learn by observing well-tracked strikes by human-made objects.
NASA hurled rocket sections and lunar modules into the moon during the Apollo era for seismic measurements.
In 2009, NASA intentionally crashed its LCROSS spacecraft and upper stage in search of ice near the lunar south pole.
The US and China are racing to land astronauts on the moon in the next few years.
Scientists said it's crucial to improve debris monitoring and traffic control before numerous robots and astronauts arrive.
Elon Musk's SpaceX and Jeff Bezos' Blue Origin are vying to provide the lander for NASA's yet-to-be-named moonwalkers of Artemis IV, who will succeed the 12 Apollo astronauts who strolled the lunar surface.
"This impact will not be a problem," retired astrophysicist Jonathan McDowell said.
"But in a future where there are long-term bases on the moon, similar impacts would be an issue and we need not to leave rocket stages in chaotic orbits of this kind."
Reservoir storage currently stands at 72%, down from 75% a week ago and nearly 10 percentage points below the long-term average for this time of year.
The reservoirs now classed as having "exceptionally low" levels are Blagdon and Wimbleball, both in Somerset, which are 56% and 54% full respectively, and Hanningfield in Essex, which is 65% full.
The seven with "notably low" levels include Ardingly in West Sussex, which is 66% full; Blithfield in Staffordshire (63%); Clatworthy in Somerset (56%); and Clywedog in Powys (69%), which provides water for central and north-west England.
Also in this category are the Dove group of reservoirs in Derbyshire (77%); Grafham in Cambridgeshire (78%); and the Lower Thames group of reservoirs in Berkshire and west London (74%).
The figures are from the Environment Agency's latest weekly dry weather and drought report.
Check the weather where you are
The Met Office said on Friday that a prolonged heatwave, exceptionally low rainfall, and widespread drought conditions have combined to make July "a remarkable month".
England and Wales are both on course for their driest July since rainfall records began in 1836.
Southern England has been particularly dry, with just 1.5mm of rainfall - which is 2% of its long-term average rainfall for the month.
Northern Ireland is on track for its eighth driest July while the UK as a whole is on course for its third driest.
Scotland has received 72% of its average rainfall for the month and is not currently within its top 10 driest Julys.
Forecasters have said temperatures in the coming days will be closer to average for this time of year.
Sky News weather producer Kirsty McCabe said the UK and Ireland can expect "cooler and fresher conditions with a few showers around, mainly in the North".