Burnham said he was "reaching out and listening" before he met Charles for the first time since his gesture to nations that endured slavery at the hands of colonial Britain.
The two men met in Antigua where the King will deliver the keynote address opening the Commonwealth Heads of Government Meeting on Tuesday, expected to be dominated by the issue of a slavery apology and reparations.
Burnham has been criticised by opponents claiming he has put the King under pressure by stating he is open to "listening to calls for a formal apology" about slavery, but ruled out reparations.
The King and Burnham were reunited at a Commonwealth Business Forum held at the Sticky Wicket, a bar and restaurant alongside a cricket ground.
As he arrived, Burnham was asked whether he had "made things more difficult" for the King.
"No. We are reaching out and listening. The whole aim here has been to come and listen and bring a new approach to the Commonwealth, and to bring people together," he said.
Britain's royal family has a long association with the transportation and selling of enslaved people for profit.
Elizabeth I backed the lucrative dealings of John Hawkins, one of Britain's first slave traders, and Charles II, encouraged the expansion of the enterprise.
As the slavery abolitionists campaigned against the trade they were opposed by the Duke of Clarence, George III's son, later to become William IV.
Burnham did not respond to a question on whether the King should make a personal apology on behalf of his ancestors.
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