"They're opening up the North Sea oil," Trump told reporters at the White House before departing for a campaign trip to Florida ahead of the November 3 midterm US elections. "I've been after them to do that for two years... Andy is getting it done."
On Wednesday, Trump praised Britain's approval of the Jackdaw North Sea gas field, a decision that ended years of legal and political wrangling.
Trump said the decision would "make the UK a very rich country".
British officials said a decision about the bigger Rosebank oilfield was yet to be made.
Burnham's office said in a statement that the two leaders agreed the Jackdaw approval would bring "significant benefits" to Britain. They also welcomed continued cooperation between British and US companies, particularly on nuclear power.
Burnham also raised the topic of oil exploration off the Falkland Islands, a largely self-governing British overseas territory whose sovereignty Argentina has long disputed.
Burnham spoke about the economic consequences for Britain and the US of what he described as Argentina's coercive measures seeking to halt the drilling and reiterated the importance of respecting the islanders' right to develop their natural resources, Downing Street added.
On Ukraine, Burnham underlined the importance of Kyiv having full access to Starlink to defend itself against Russia and welcomed Trump's efforts to bring Russian President Vladimir Putin to the negotiating table. The leaders agreed to work with international partners to end the bloodshed, the statement said.
The leaders also discussed the Middle East, with Burnham reaffirming Britain's support for Saudi Arabia in the face of Houthi attacks.
