Mr Trump predicted he would have a “fantastic” relationship with the Prime Minister, as he welcomed her as his first overseas visitor since becoming president and accepted her invitation to come to the UK on a state visit later this year.
In a White House press conference – which was held minutes after he showed her the bust of Winston Churchill which has been restored to a space in the Oval Office – Mr Trump said the special relationship between the UK and US was “one of the great forces in history for justice and for peace”.
And he added, in words that will be warmly appreciated in Downing Street: “We pledge our lasting support to this most special relationship.”
The sense that Mrs May had hit it off with the president on their first meeting was reinforced when the pair briefly held hands as they walked from the Oval Office to their first press conference together.
In almost an hour of talks ahead of a working lunch, Mrs May appeared to have made some progress on key policy issues which have threatened to divide the two.
She pointedly noted that she had secured Mr Trump’s “100 per cent” commitment to Nato, allaying British concerns over his earlier description of the military alliance as “obsolete”.
Addressing Mr Trump directly in front of the TV cameras, Mrs May said: “Today we have reaffirmed our unshakeable commitment to this alliance. Mr President, I think you confirmed that you are 100 per cent behind Nato.”
And Mr Trump backed away from suggestions that he was ready to sanction the use of torture on terrorism suspects – something Mrs May had previously made clear she could not support.
The president said that, although he did not “necessarily agree” with his defence secretary’s opposition to “enhanced interrogation” methods such as torture, he would allow James Mattis to override him.
High on the agenda for the meeting were Britain’s hopes for a swiftly-organised free trade agreement with the US after the UK’s withdrawal from the European Union.
Mrs May said she and Mr Trump were both “ambitious” for a deal and wanted to “take forward immediate high-level talks, lay the groundwork for a UK-US trade agreement and identify the practical steps we can take now in order to enable companies in both countries to trade and do business with one another more easily”.
Mr Trump left no doubt about his enthusiasm for the process.
He said: “I think Brexit is going to be a wonderful thing for your country.
“When it irons out you are going to have your own identity, and you are going to have the people that you want in your country, and you are going to be able to make free trade deals without having somebody watching you and what you are doing.
“I think it will end up being a fantastic thing for the United Kingdom. I think in the end it will be a tremendous asset, not a tremendous liability.”
Speaking at his first press conference since his inauguration last week, Mr Trump seemed taken aback when confronted by a tough series of questions on torture, Russia, travel restrictions for Muslims and abortion.
Raising his eyebrows, he asked Mrs May: “This was your choice of a question? There goes that relationship.”
Mrs May said: “I have been listening to the president and the president has been listening to me. That’s the point of having a conversation and a dialogue.”
She added: “There will be times when we disagree and issues on which we disagree. The point of the special relationship is that we are able to have that open and frank discussion so we are able to make that clear when it happens.
“But I am clear also that there are many issues on which the UK and the US stand alongside one another, many issues on which we agree.”
She said an “even stronger special relationship” would be in the interests of the wider world.
The press conference came after Mr Trump’s senior adviser said US sanctions against Russia and other issues would be on the table when the president talks with Russian president Vladimir Putin today.
Kellyanne Conway said that Mr Trump would be receptive if Mr Putin wants to have a serious conversation about how to defeat Islamic extremists.
Barack Obama’s administration and the European Union imposed sanctions on Moscow for its annexation of Ukraine’s Crimean peninsula and support for a pro-Russia insurgency in eastern Ukraine.
Relations have plunged to post-Cold War lows over Ukraine, Mr Putin’s backing of Syrian president Bashar al-Assad and allegations of Russian interference in the US elections.
In a separate development, Mr Trump yesterday sent a message of support to anti-abortion activists holding a large rally in Washington DC.