Charles will preside over the Commonwealth Heads of Government Meeting in his role as the head of the "family of nations" after travelling from a state visit to Guyana.
The King will give the opening address to assembled presidents, prime ministers and other national representatives of the 56 member states on Tuesday.
His speech is likely to reflect the change in the government's position on the legacy of slavery.
Before the trip, it was understood ministers had not advised the King to offer an apology or reparations, in keeping with established policy.
But the prospect of a formal expression of regret is now closer given the prime minister's comments in The Guardian on Friday.
"We are offering a structured dialogue with those affected by the transatlantic slave trade," Mr Burnham wrote.
"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."
The PM expressed his "sorrow and regret for the pain and suffering of so many millions" but ruled out considering financial reparations.
"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," he said.
Caribbean nations made renewed calls for Britain to apologise for its role in selling enslaved people for profit, and for reparations, in the run-up to the King's tour of the Caribbean.
Before departing Guyana on Friday, the King visited Independence Park in the capital city of Georgetown, where 33 people involved in the Demerara Rebellion of 1823 were hanged.
At the park, Charles was joined by Guyana's culture minister Steven Jacobs to acknowledge the country's history of slavery and resistance.
Charles began his nine-day trip last Tuesday by acknowledging the "great suffering of the past" in a speech at a youth rally in the Bahamas.
Ahead of the visit, Guyana's President Mohamed Irfaan Ali said Charles had recognised slavery as the darkest days in human history, but an apology would be "the next step".
On Friday, about 15 people protested ahead of the king's arrival in Georgetown, Guyana. They gathered outside the Umana Yana, a conical palm-thatched hut originally built for the 1972 Non-Aligned Foreign Ministers Conference, where Charles later delivered a speech.
"It's an opportunity to try to impress upon him this historic responsibility for what took place here," said Terence Campbell, a member of parliament for the opposition APNU, who stood among the protesters.
From the 15th to the 19th century, at least 12.5 million Africans were abducted and sold into slavery by European traders, with Britain transporting an estimated 3.2 million people across the Atlantic, making it one of the biggest slave-trading nations, according to the Transatlantic Slave Trade Database.
With PA
