The prime minister is due to attend the Commonwealth Heads of Government Meeting (CHOGM) in Antigua and Barbuda - as is the King who, as the head of the "family of nations", will preside over the event.
Charles will give the opening address to national representatives of the 56 member states on Tuesday.
His speech is likely to echo the government's changed position on the legacy of slavery, the Press Association reports.
Ahead of his nine-day trip, it was understood ministers had not advised the King to offer an apology or reparations, in keeping with established policy.
But a formal apology is now potentially closer following comments made by Mr Burnham to the Guardian on Friday.
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The prime minister wrote: "We are offering a structured dialogue with those affected by the transatlantic slave trade.
"Listening to calls for a formal apology on behalf of the British state will form an important part of this process and I am open to them."
Mr Burnham expressed his "sorrow and regret for the pain and suffering of so many millions" but ruled out financial reparations.
He wrote: "Some have called for financial reparations for the slave trade. I do not believe that is the right answer.
"I do not think that British people today can be held liable for historical wrongs, however heinous, that happened in a different age under different laws and in which they played no personal part."
The prime minister of Barbados, Mia Mottley, said in a post on X: "Barbados welcomes the UK prime minister's acknowledgement of the brutality of the transatlantic slave trade, his openness to a formal apology and his offer of structured dialogue."
Jamaica's prime minister, Andrew Holness, said of Mr Burnham's comments: "Open-minded dialogue is always a good place to start addressing difficult issues."
Caribbean nations again urged Britain to apologise for its role in selling enslaved people, and for reparations, in the lead-up to the King's Caribbean tour.
Before leaving Guyana on Friday, the King visited Independence Park in the capital, Georgetown, where 33 people involved in the Demerara Rebellion of 1823 were hanged.
He was joined by Guyana's culture minister Steven Jacobs to acknowledge the country's history of slavery and resistance.
Charles started his Caribbean trip last Tuesday with a speech at a youth rally in the Bahamas, where he commented on the "great suffering of the past".
Prior to the visit, Guyana's President Mohamed Irfaan Ali said the King had recognised slavery as the darkest moments in human history, but an apology would be "the next step".
Sky News has learnt that VMO2, which is jointly owned by Liberty Global and Telefonica, is drawing up plans to make a substantial proportion of its 16,000-strong workforce redundant in the coming months.
Sources close to the company said precise numbers had still to be finalised ahead of its third-quarter results next Tuesday, when a cost-reduction target will be confirmed in financial - rather than headcount - terms.
They conceded, though, that layoffs at VMO2 would be in the thousands, with one insider suggesting as many as 4,000 jobs were likely to be affected.
Trade union officials at the CWU and Prospect are understood to have been briefed on some of the impending transformation plans.
The majority of the affected staff will be those in non-customer facing roles, according to one insider, while a number of senior leadership changes are already under way at the company led by chief executive Lutz Schuler.
VMO2 is also expected to review its use of contractors and suppliers in a bid to bring down costs.
"The final numbers on cost reduction are still moving around," said one person close to the situation.
A VMO2 spokesman declined to comment on the redundancy plans.
The quarterly results will be presented against a backdrop of mounting fears about the company's debt pile, with the price of its bonds falling sharply during the summer.
Next week's update is likely to point towards a broader strategic transformation of VMO2's business at a time of lacklustre growth in the UK economy.
The Financial Times reported in July that VMO2 was considering options to cut its indebtedness, including slashing its dividend and capital expenditure.
When confirmed, the job losses at VMO2 will add to significant layoffs at BT Group and Vodafone Three as telecoms operators seek to become more efficient because of rising cost pressures.
Sky, the immediate parent of Sky News and a competitor to VMO2 in the provision of broadband and pay-television services, also cut hundreds of jobs in September last year.
The looming restructuring at VMO2 comes amid a shake-up of Britain's telecoms market being driven by competition policymakers and the recent collapse of TalkTalk Telecom Group.
BT's emergency takeover of TalkTalk has angered rivals fearful that it will entrench the market power of the former state monopoly and its Openreach infrastructure division.
The FT reported this week that VMO2 was nearing a deal to use Openreach's network in parts of the country to extend its footprint from 19 million to as many as the 25 million homes covered by Openreach.
VMO2 is also expected to halt its own build-out programme to reduce its capex budget.
A number of fibre altnets have collapsed in the last two years because of the industry's deteriorating economic dynamics.
CityFibre's efforts to become the most significant rival to Openreach have been stymied by a large debt pile of its own, which it is now in talks to restructure by asking some lenders to take a significant haircut.
The Competition and Markets Authority (CMA), which was asked to fast-track a review of BT's TalkTalk deal, has also sparked anger at VMO2 by signalling that it may block a £2bn takeover of Netomnia by nexfibre, a wholesale fibre joint venture whose participants include both of VMO2's shareholders.
Instead, the CMA said its preference would be for CityFibre to acquire Netomnia in order to create a stronger challenger to Openreach.
Its statement is being vigorously opposed by Netomnia's own shareholders.
A source close to one said that BT's 6.7% share price surge on Friday - on the back of the news about Openreach being used by VMO2 - "confirms what we have been saying, that Ofcom's intent to create competition died when they went against the [nexfibre-Netomnia] merger".
"The logic of Ofcom and the CMA is not rooted in reality.
"Hopefully they will realise that in time to prevent more damage to competition in the UK market."
Sources said that Netomnia's shareholders were scheduled to meet both regulators next week to protest at the unfolding situation.
The Hollywood actress, 54, said she was "so sorry" to "the people that I disappointed, that I hurt, I frightened, that I alarmed".
In a video on Instagram on Friday, the Cruel Intentions and Legally Blonde star said that even though her lawyers advised her against talking about the case, she was taking the matter "very seriously".
She has pleaded not guilty to state charges over allegations she kicked a police officer and a nurse after a mid-air incident involving crew members on 11 October.
Blair was on a Delta Air Lines flight from Los Angeles to New York City when she allegedly became "physically abusive" and was removed from the plane, authorities said at the time.
Two flight attendants reported injuries, including one who was taken to hospital, the Port Authority said.
Blair, who was diagnosed with multiple sclerosis (MS) in 2018, was arrested when the flight landed in New York and transported to hospital for evaluation.
A police officer who took her there alleged that while securing Blair to a stretcher, the actress kicked her repeatedly around the face, neck and chest, causing serious pain and substantial swelling.
And a nurse moving Blair said the star kicked her in the chest.
She was charged with two counts of second-degree assault at a court in Queens and then released on bail, as a judge set the next hearing for January.
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In her Instagram video, Blair said: "I know I've been quiet for a couple weeks since October 11. And you deserve to hear from me directly, and I'm sorry that it's taken this long.
"There is a legal matter going on, and my lawyers have asked me to not say anything about that day, and I'm listening to them."
The actress said she took the matter "very seriously" and the "very simple and the truest thing I know about myself is that not one day ever have I intended to hurt or frighten anyone".
"And so to the people that I disappointed, that I hurt, I frightened, that I alarmed, I am so sorry," she said.
She thanked those who reached out with love and support, and then assured fans her service dog Scout, who accompanied her on the flight, was doing well.
She got the Labrador five years ago after her MS diagnosis.
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Blair's lawyer Shawn Barnes said in court that her actions were the result of a medical emergency, citing diagnoses of multiple sclerosis and Parkinsonism.
At the 12 October arraignment hearing, Mr Barnes told the judge that her actions were not intentional and that the star had missed several doses of medication and also fell and hit her head on the flight.
Blair's representative separately said Blair had "experienced a medical emergency that led to the alleged conduct".
Around 10,000 people been infected since May, making it the biggest-ever outbreak of the disease in DRC.
The number of deaths has increased by about 1,000 since the start of October.
There are so far no approved treatments or vaccines for the rare Bundibugyo species responsible for the outbreak.
Fears have been growing over the epidemic spreading to nearby countries. Nigeria last week became the fourth African nation to record a case after DRC, Kenya and Uganda.
Eight of DRC's 26 provinces have recorded cases, but the response has been blighted by a series of attacks on health workers.
A nurse was killed and another health worker was injured last week by a group of young people with machetes and batons.
Misinformation has been blamed for the attacks, and authorities said the attackers had accused the pair of spreading the disease by contaminating water sources.
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The house of another community worker raising awareness about Ebola was set on fire last week, officials said, while a politician was beaten to death last month for defending measures in dealing with the outbreak.
The DRC outbreak is now also the world's second-largest ever; only behind the 11,000 who died and 28,000 infected between 2014-2016 in Guinea, Liberia and Sierra Leone.
Warning - contains details of injuries some may find distressing
Simon Richardson, 35, failed to give way at a stop sign in Greater Manchester and struck headteacher Michael Mullins.
The father-of-two, who was travelling home from Stretford Grammar School, hit the bonnet and windscreen before going underneath the Volvo on the evening of 20 April.
Prosecutors said Richardson chose to drive on from the scene in Paddock Lane, Warburton, despite knowing Mr Mullins was trapped inches beneath his feet.
Phil Barnes KC told jurors: "Simon Richardson knew he was there and he inevitably knew what the consequences of driving on would be, and he did it anyway."
Richardson's cousin and front seat passenger, Gabrielle Debotte, described hearing "moaning" and said Richardson had told her: "He's under the car."
The killer drove 1.3 miles to a farm track and twice tried to pull Mr Mullins out.
Jurors at Manchester Minshull Street Crown Court heard he was unsuccessful so got back in and swerved sharply from side to side to try to dislodge him.
Mr Mullins was eventually released, but Richardson drove over him again as he reversed and fled.
The court was told a motorcyclist, who had followed the trail of blood, found the 56-year-old on a grass verge. Emergency services were called but he could not be saved.
Richardson, of no fixed address, later changed vehicles to try to avoid capture.
He was arrested the next day after a police chase.
Richardson claimed he had not realised Mr Mullins was under the car and denied getting out to try to free him.
The jury did not believe him and he will be sentenced for murder - as well as dangerous driving over the police chase - on 2 November.
The family of Mr Mullins - who was from Northwich, Cheshire - were in tears on Friday as the verdicts were read out.
His wife, who asked not to be named, said in a statement: "My past, present and future have been brutally and callously torn apart.
"My memories of being with Michael through the rough and the smooth are now permanently tainted with sadness.
"Michael is irreplaceable to family, friends and the school."
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In their own tributes, his son and daughter both described him as a loving and devoted father who was always there for them.
"I am here telling you all that my dad was and always will be our everything," his daughter said.
Mr Mullins' son said he "was the type of dad that would show up to everything".
He added: "Our last exchange of words together before he dropped me at the station was him reminding me of how proud he was of me."