The pair spoke over the phone on Monday, within an hour of Mr Burnham arriving in Downing Street.
Speaking to the BBC’s Sunday with Laura Kuenssberg programme, Mr Burnham said: “I can’t at any point say that I won’t take a different opinion from him [Donald Trump], that I will need to voice a different idea that is right for Britain.”
The new Prime Minister said he would publicly disagree with Mr Trump if necessary, adding: “You have to defend your own national interest before anything else. That’s what you’re required to do if you’re to do this job properly.”
After the call, Mr Trump stated in a post on Truth Social that Mr Burnham “has got a big job ahead of him, but he will be able to do it” and pledged that the US “will be there to help”.
Downing Street described the 20-minute conversation as “extremely warm”.
It comes after relations between Washington and Sir Keir Starmer’s government became increasingly fractious over the Iran conflict.
Speaking to the public broadcaster, Mr Burnham ruled out an early general election, saying Britain had “voted for a manifesto” in 2024.
“There’s no early general election. I don’t think people want it,” he said.
“I just think what I most need to do is get Britain focused on getting the country back where it should be.”
The PM also refused to be drawn on a timetable for increasing defence spending to 3% of GDP.
Mr Burnham said he was “very aware” of Chancellor John Healey’s position and resignation as Sir Keir’s defence secretary following disagreements over the issue.
He said: “The first challenge facing us both is to ensure that the defence investment plan is fully funded and that’s the thing that’s right in front of us and we need to work that through as we go towards the budget later this year.”
Elsewhere, new Defence Secretary Wes Streeting said he shared “very warm conversation” with his US counterpart Pete Hegseth.
The pair are thought to have brainstormed ideas for an international conference which allies of the UK and US would attend, but have not yet firmed up the details of a summit.
A source close to Mr Streeting last week said: “The Defence Secretary had a very warm conversation with Secretary Hegseth.
“They discussed the situation in the Strait of Hormuz and how allies can work together.
“They also discussed defence investment and the UK’s commitment to meeting our Nato commitments.”
