Trump's trip coincides with a tumultuous week for May after two senior ministers resigned in protest at her plans for trade with the European Union after Britain leaves next March.
Standing next to May during a joint news conference at the British Prime Minister's official country house, Chequers, Trump announced he had at least started to apologize earlier in the morning.
Trump's interview with The Sun was full of provocative statements.
"There is no stronger alliance than that of our special relationship with the U.S. and there will be no alliance more important in the years ahead", the British leader said. Foreign Secretary Boris Johnson and Brexit Secretary David Davis both quit the government this week in protest.
Both Trump and May dismissed the idea that the United Kingdom government's plan makes a trade deal impossible.
He is due to meet the Queen later today at Windsor Castle before flying to Scotland, where he will play golf at one of the courses he owns, in Turnberry.
Our Brexit Insider Facebook group is the best place for up-to-date news and analysis about Britain's departure from the European Union, direct from Business Insider's political reporters. "I think he's got what it takes, and I think he's got the right attitude to be a great prime minister".
Trump told the newspaper he had advised May on how to conduct Brexit negotiations, "but she didn't listen to me".
Britons are narrowly in favour of US President Donald Trump airing his views on Brexit, a Sky Data poll suggests. I get along with her very nicely.
Think your friends would be interested? However, that has angered hardliners in May's Cabinet and prompted high-level resignations in the past week, including the departure of Foreign Secretary Boris Johnson, who Trump called "a very talented guy".
"I was very saddened to see he was leaving government and I hope he goes back in at some point".
"They all showed up on the 9th hole, overlooking the ocean, and I said what's going on, and all they wanted to talk about was Brexit".
The mayor of London defended his decision to allow protesters to fly a giant "Trump baby" blimp over the city during President Donald Trump's visit, saying disagreement was part of a fundamental freedom.
"I said, 'I want to apologise because I said such good things about you, ' And she said, 'Don't worry, it's only the press, '" Trump said of their purported conversation.
"I look at cities in Europe, and I can be specific if you'd like". I haven't been there in a long time.
"If you reported them that's good", Mr Trump told the political correspondent.
Trump's schedule, however, will largely keep him out of central London.
There are reportedly more than 60 protests scheduled to take place throughout Trump's four day working visit to the nation, with roughly 50 to 100,000 people expected to come out to oppose the president, according to CNBC.