The BBC’s political editor, Laura Kuenssberg, said Mr Trump’s interview had “driven a bulldozer” through Mrs May’s claim that the UK would be able to get decent trade deals with the wider world, while sticking to the EU rules.
Mr Trump’s interview has provoked strong reaction.
Responding to Mr Trump’s criticism of his response to terrorism, Mayor of London Sadiq Khan said it was “interesting” that the US president “is not criticising the mayors of other cities” which have also experienced terror attacks.
He defended his decision to allow the giant Trump baby inflatable to fly over London, saying: “The idea that we limit the right to protest because it might cause offence to a foreign leader is a slippery slope”.
Conservative MP Sarah Wollaston called Mr Trump’s comments “divisive, dog-whistle rhetoric”, and added: “If signing up to the Trump world view is the price of a deal, it’s not worth paying.”
Labour MP Phil Wilson, supporter of the People’s Vote campaign, which calls for a public vote on the final Brexit deal, said Mr Trump’s comments showed the prime minister’s “botched” Brexit proposal was “worsening hour by hour”.
He said Mr Trump had “explicitly backed those who advocate a disastrous no-deal Brexit.”
Mr Trump’s comments came on the same day the UK government published its proposal for its long-term relationship with the EU.
The long-awaited plan is aimed at ensuring trade co-operation, with no hard border for Northern Ireland, and global trade deals for the UK.
But leading Brexiteers Boris Johnson and David Davis resigned from the cabinet days after ministers reached agreement on the plan at Chequers a week ago.
Responding to an earlier suggestion by President Trump that the British people were not getting the Brexit they voted for, Mrs May said: “We have come to an agreement on the proposal we are putting to the European Union which absolutely delivers on the Brexit we voted for.”
Analysis, by BBC political editor Laura Kuenssberg
President Trump drives a bulldozer through the government’s central claims about their compromise – that the UK would be able to get decent trade deals with the wider world, while sticking to the EU rules.
A lot of this visit has been carefully choreographed, as the prime minister and the president dance around each other.
But if the president really wanted to help her build support for her controversial compromise, this isn’t the way to do it.
At Thursday’s dinner, Mrs May said that more than one million Americans work for UK-owned firms, telling Mr Trump: “As we prepare to leave the European Union, we have an unprecedented opportunity to do more.
“It’s an opportunity to reach a free trade agreement that creates jobs and growth here in the UK and right across the United States.”
As Mr Trump arrived in the UK, protesters gathered outside the US ambassador’s residence in in Regent’s Park, London, and an estimated 1,000 of them demonstrated near Blenheim Palace itself, the birthplace of wartime Prime Minister Winston Churchill.
John Rees of the Stop the War group addressed protesters, saying of Trump: “He’s a wrecking ball for race relations, he’s a wrecking ball for prosperity, he’s a wrecking ball for women’s rights, he’s a wrecking ball for any peace and justice in this world and we have to stop him.”
On Friday, Mrs May and Mr Trump will go to watch a joint counter-terrorism exercise by British and US special forces at a military base.
The pair will then travel to Chequers – the PM’s country residence in Buckinghamshire – for talks with Foreign Secretary Jeremy Hunt.
Extra security is in place to police protests planned for the second day of Mr Trump’s visit.
The president and first lady will travel to Windsor on Friday afternoon to meet the Queen, before flying to Scotland to spend the weekend at Mr Trump’s Turnberry golf resort. This part of the visit is being considered private.