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".
Earlier, the Constitutional Tribunal said its longest-serving judge Anna Korwin-Piotrowska had suspended Bogdan Swieczkowski, an appointee of Prime Minister Donald Tusk's far-right predecessor, and was now in charge of the court.
President Karol Nawrocki turned back to Poland while en route to Senegal as the row over the suspension of his far-right ally intensified, escalating a battle with Mr Tusk's government over control of the judiciary.
"At a moment like this, the President must be in Poland," Mr Nawrocki said on X hours before he was pictured arriving at the court in Warsaw.
He later met the first president of the Supreme Court, Zbigniew Kapinski, at the city's Presidential Palace.
Police said they detained three people after some protesters breached the tribunal's fence and others threw objects at officers.
Under a heavy police presence, right-wing politicians addressed the crowd, accusing Mr Tusk's government of staging a coup against the court.
Mr Tusk said in a post on X that his government was seeking to restore the rule of law, which the European Union had accused his predecessors - the Law and Justice Party (PiS) - of subverting.
"This process is difficult, at times even painful, and requires decisive action. There is no other way," he said.
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A long-running political dispute over the Constitutional Tribunal has destabilised the institution, with the pro-European government and the nationalist president refusing to recognise appointees put forward by the other side.
Politicians from PiS, currently the largest opposition force, criticised Mr Swieczkowski's suspension, with former justice minister Zbigniew Ziobro calling it a "blatant, criminal attack on the constitutional order".
"That is why I call for resistance that will be loud, firm, solidary and fearless," he wrote on X.
Pawel Soloch, ambassador of the Republic of Poland in Romania, announced his resignation after the suspension.
He said in a post on X: "In connection with the fact of an attempt to unlawfully take over the Constitutional Tribunal, I see no possibility of continuing to serve as the Ambassador of the Republic of Poland in Romania."
Meanwhile, the US ambassador to Poland, Tom Rose, said: "The events unfolding tonight at Poland's Constitutional Tribunal are very serious indeed. We appear to be witnessing an unprecedented confrontation over the government's use of state power and the independence and authority of a constitutional institution."
He added: "America is on Poland's side. It is not our place to judge or interfere in its internal affairs. Our concern is for the stability, security, and democratic future of a country we are proud to call our friend and ally."
Mr Swieczkowski became president of the Constitutional Tribunal under PiS, which came to power in 2015 and brought the courts under its judicial control.
Mr Tusk, whose centre-left government took over from the party in December 2023, has accused Mr Swieczkowski of blocking judges elected by the parliamentary majority from taking up their posts at the tribunal.
Discussing the future of the tribunal at Campus Polska in August, Mr Tusk said his government had a plan to ensure the institution would again be staffed with lawfully appointed judges, while acknowledging his methods may not please everyone.
In 2015, the nationalist PiS sought to appoint judges to seats already filled by the previous government.
Its nominees gained a majority on the Constitutional Tribunal, while critics at home and abroad said the court no longer met the standards of an independent and impartial tribunal established by law.
After Mr Tusk's coalition took office in 2023, it vowed to restore the rule of law and reform the court, but PiS ally Mr Nawrocki has acted to crush those efforts since his election as president in 2025.
Earlier this year, Mr Nawrocki refused to swear in three of six judges nominated by parliament, while Mr Swieczkowski, a former deputy justice minister in the PiS administration, declined to seat judges elected by the current parliamentary majority.
Last month, the parliament appointed Mr Tusk's ally Maciej Berek as a judge to the tribunal, which critics said undermined the government's stated aim of restoring checks and balances.
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."
The disgraced US financier visited Balmoral with Ghislaine Maxwell in 1999, reportedly on the invitation of the then-Prince Andrew.
Police Scotland said it had received information about time the royal spent with the late paedophile at the King's Highlands estate.
These inquiries relate to claims that women were trafficked to the Scottish castle, according to the Daily Express.
A spokesperson for the force said: "We have received information and active enquiries are ongoing but these are at a very early stage.
"All enquiries are evidence led, with the needs of victim survivors at the heart of those enquiries. We would never confirm details of ongoing enquiries and it would be inappropriate to comment further."
Pictures previously released by the Department of Justice in the US appeared to show Andrew and Epstein on a shooting trip on the grounds of the sprawling residence.
Multiple forces in the UK, including Thames Valley Police, the Metropolitan Police and Surrey Police, also have live investigations linked to the release of the Epstein files, the National Police Chiefs' Council (NPCC) said.
Located in the Scottish Highlands, Balmoral Castle has belonged to the Royal Family since 1852 when Queen Victoria and Prince Albert acquired the estate.
The castle has been described as "a treasured retreat for the Royal Family" and was where Queen Elizabeth II died at the age of 96 in 2022.
King Charles stripped his brother of both his right to be a prince and his dukedom last year over his association with convicted sex offender Epstein.
The former Duke of York was arrested on suspicion of misconduct in public office on 19 February, his 66th birthday, following allegations he shared sensitive information with Epstein during his time as the UK's trade envoy.
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Andrew was interviewed under caution before being released under investigation, with the police probe ongoing.
He has always denied any wrongdoing.
Andrew was the UK's special representative for international trade and investment from 2001 until 2011, when he stepped down amid controversy over his friendship with Epstein.
Virginia Giuffre, who said she was trafficked to the former duke at the home of convicted sex trafficker Maxwell, was paid millions of pounds by Andrew to settle a civil suit in the US in 2022.
Ms Giuffre died by suicide aged 41 in April 2025.