It is not yet known whether they will join the duke on any visits outside the capital.
Prince Harry had intended to travel with Meghan and their children, Archie and Lilibet, to an event marking one year until the Invictus Games in Birmingham.
He's also due to visit some charities he supports.
It has been four years since the children last saw their grandfather in person and the King has offered to put the family up at an unnamed royal residence.
There have been concerns over how they would be protected in the UK, amid reports last weekend that a request for taxpayer-funded police protection had been denied.
At the time, a source said the duke could decide to leave his family at home.
"The hope is they can meet their grandfather," the source added. "But there is no way that can happen if they are chased by paparazzi wherever they go from the moment they step off the plane."
However, it emerged yesterday that the prince's security team was still looking at options to make the visit viable for the whole family.
The duke last year lost a legal battle with the government over his security in the UK after it was downgraded when he decided to stop being a working royal in 2020.
He is still waiting for Ravec, the committee that decides on protection for royalty and public figures, to review his case after making a personal appeal in December.
The reason for Harry not bringing his family to London is likely due to concerns from his security team and the lack of an up-to-date assessment by the Risk Management Board (RMB), whose work informs Ravec.
It was meant to have met in March and would have given a fuller picture of the potential threats faced by the duke and his family, but the meeting didn't go ahead.
Archie, seven, and Lilibet, five, last saw the King at the late Queen's platinum jubilee celebrations in 2022. Sources stress the monarch is not involved in decisions about his son's security.
Read more from Sky News:
Boy hurt in crocodile incident has had five surgeries
Evri suing BBC over Panorama programme
When Harry lost his court appeal last year, he told the BBC: "I can't see a world in which I would be bringing my wife and children back to the UK at this point and the things that they're going to miss is, well, everything you know."
A government spokesperson said: "It is our long-standing policy not to provide detailed information on protective security arrangements, as doing so could compromise their integrity and affect individuals' security."
Peak temperatures of 34C are forecast and a new hosepipe ban has been announced.
Temperatures in the South of England could reach 28C on Saturday, according to the Met Office.
It said temperatures are forecast to rise to 29C on Sunday, then approach the low 30s next week, before the South East sees a peak of 34C on Thursday or Friday.
It is expected to be a prolonged spell of hot weather in parts of the UK, but the forecaster said it will not be as hot and humid as last month's heatwave.
See the weather forecast for your area
Yellow health alerts issued by the UK Health Security Agency (UKHSA) come into force at midday in parts of the UK, with health officials cautioning that high temperatures over the coming days could cause a greater risk to life for vulnerable people.
The warnings cover the East Midlands, east, southeast and southwest of England, as well as the West Midlands, until 8pm on 11 July.
The agency said minor impacts were likely across health and social care because of the high temperatures within the warning areas, including increased use of healthcare services by vulnerable people and a greater risk to life to the same group.
It also warned of a potential increase in water-related deaths, including risks from cold water shock and drowning.
Meanwhile, Southern Water has issued a hosepipe ban from 12.01am on 10 July in Hampshire and on the Isle of Wight, while South East Water's temporary hosepipe ban for people living in Kent came into force on Friday.
The bans stop customers using their hosepipes for non-essential activities, including watering gardens, using a sprinkler system, filling paddling pools, hot tubs, swimming pools, ponds or fountains, and washing cars, patios or windows.
Read more from Sky News:
Boy attacked by crocodile has surgeries
Funeral of Iran's supreme leader
Met Office meteorologist Becky Mitchell said: "We have got a heatwave on the way across parts of southern England and Wales, temperatures are already quite high across the South East today, we could get 28C in London.
"Elsewhere, it's not as hot as we have got temperatures closer to average but they will climb as we go into next week.
"The northern areas of the country will be cloudier with spells of rain at times, so there's a north/south split.
"There will be highs of 29C tomorrow in the south. At the start of next week we will see temperatures approach the low 30s."
She added: "The peak of the heat appears to be Thursday or Friday, 34C in the South East.
"By comparison to the heatwave we saw in June, this heatwave won't be as hot and humid, but it will be a prolonged spell of hot weather which lasts around a week."
Last week the UK set a provisional June temperature record of 37.7C in Lingwood in Norfolk, according to the Met Office.
It beat the previous June record for the UK of 35.6C, which dates back to 1976.
The parents said they have been "living at the hospital" and their son faces more surgeries after he was attacked by the reptile at Johnsons of Old Hurst in Cambridgeshire on 18 June.
A fundraising page set up to support the boy's recovery and rehabilitation has now raised more than £25,000.
It will also provide the family with financial help while they support him in hospital.
In an update, his parents said: "We are extremely grateful for everyone's generosity.
"We could never thank you enough for the support you have given our family in this horrible time."
They added: "We have been living at the hospital for two weeks now and our son has undergone five surgeries."
He will have to undergo at least two more surgeries and the last two weeks have been "a very uncertain and unsettling time", his parents added.
The update continued: "His rehabilitation journey is uncertain but we do know that it won't be a short process. The money that has been donated so far has been used to pay for our expenses.
"We are both currently away from work and the time in which this will need to be the case is unknown."
In the statement, the parents added: "The donations have helped ease the pressure of these expenses so we are extremely grateful to everyone.
"Further donations will be used to support our son's rehabilitation and to give back to the hospital who have helped us in many ways during this challenging time. Thank you."
A 30-year-old man from Norfolk was arrested on suspicion of attempted murder when police arrived at the scene.
He was later bailed as officers said he was "assessed as not being fit for interview".
Police said the boy remains in Addenbrooke's Hospital in Cambridge in a "stable condition", having been "injured by a crocodile" after ending up in the enclosure.
The Tropical House where the crocodiles are kept was closed following the incident, but reopened last week.
On Friday the owners of the zoo praised the "incredible speed and determination" of those who helped the boy.
Read more from Sky News:
Funeral of Iran's supreme leader
Evri suing BBC for £1.2m
In a social media post, Johnsons of Old Hurst said they were "alerted to reports that a child had allegedly been thrown into one of our crocodile enclosures" on 18 June.
"Everyone who was on site that day acted with incredible speed and determination," the Instagram post said.
"Within minutes of the first radio call, the child had been rescued from the enclosure and was receiving first aid.
"The emergency services arrived very quickly, supported by the MAGPAS Air Ambulance, and we cannot thank them enough for their professionalism, compassion and dedication in such difficult circumstances."
Johnsons of Old Hurst expressed "heartfelt thanks to the visitors who stepped forward to help, including off-duty paramedics and police officers".
They said their "calm and immediate actions made a real difference".
Earlier in the week, the boy's family thanked staff at the zoo who rescued their son.
The zoo's co-owner, Tracey Johnson, reportedly jumped into the enclosure to save their child.
Meanwhile, an investigation has been launched after the medical records of the boy were accessed by around 40 members of hospital staff.
Next Thursday: We examine a claim made by several readers this week that the UK has the least generous state pension in Europe.
Plus, a tip to get free mobile data abroad.
Every week, subscribers get early access to our Money Problem feature too and, in our popular Money in your pocket section, a rundown of the best deals currently on the market for broadband, mortgages, savings, bank switching and energy.
So join our growing Money community - and thanks to the hundreds of thousands of you who already have.
The person was in the water on Friday at Jones Beach State Park Field 6 when they suffered lacerations on their foot, officials said.
Lifeguards responded immediately and the swimmer was taken by ambulance to hospital, where they were treated for non-life-threatening injuries, they said.
Swimming at the beach where the attack happened was suspended as a search was mounted for sharks and other potentially dangerous marine life.
After an hour of searching, none were found and swimming was allowed again - though swimmers were restricted to wading up to their waists.
Read more from Sky News:
Evri suing BBC for £1.2m over Panorama programme
The most striking pictures of the week
The attack comes after shark sightings were reported at other beaches in New York state, in Queens and Long Island.
New York City officials on Thursday reported multiple shark sightings at Rockaway Beach in Queens and imposed intermittent beach closures.
New York City Emergency Management also issued a warning advising of the closures and urging beachgoers to follow the guidance from lifeguards.
Lifeguards at a Long Island beach also spotted a shark on Thursday in the water off Point Lookout and immediately ordered people out of the ocean.
Point Lookout and neighbouring Hempstead beaches were also temporarily closed to swimming after the sighting.
While sharks are commonly found in the waters off the US, shark bites are rare, experts say.