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Theresa May and Donald Trump are holding talks at her country retreat Chequers, following his controversial comments on the PM's Brexit plan.
In an interview with the Sun, Mr Trump said the PM's plan would "probably kill" any trade deal with the US.
But on Friday, he said he and Mrs May had "probably never developed a better relationship" than during this trip - his first to the UK as president.
Meanwhile, a giant blimp of Mr Trump as a baby is floating in central London.
It is part of a demonstration in Parliament Square, one of many due to take place across the UK on Friday.
In his interview with the Sun, Mr Trump also said that former Foreign Secretary Boris Johnson - who disagrees with the PM on Brexit and resigned this week - would make a "great prime minister", adding "I think he's got what it takes".
He also renewed his criticism of London Mayor Sadiq Khan over last year's terror attacks in London, saying he had done "a terrible job".
Downing Street has not yet reacted to Mr Trump's remarks, but Chancellor Philip Hammond said the talks will be "very positive".
Theresa May has been making the case for a US free trade deal, and says Brexit is an "unprecedented opportunity" to create jobs in the UK and US.
The US president and his wife, Melania, were given a red carpet reception at Blenheim Palace, Oxfordshire on Thursday evening.
They were at a black-tie dinner with Mrs May as news broke of his interview with the newspaper, which said it was conducted while he was in Brussels.
After it was published, White House spokeswoman Sarah Sanders said the president "likes and respects Prime Minister May very much", adding that he had "never said anything bad about her".
Protesters near Blenheim Palace making their feelings known on Thursday
Mr Trump - who has been a long-time supporter of Brexit - told The Sun that the UK's blueprint for its post-Brexit relations with the EU was "a much different deal than the people voted on".
He said the Brexit proposals Mrs May and her cabinet thrashed out at the PM's country house Chequers last week "would probably end a major trade relationship with the United States."
"We have enough difficulty with the European Union," he said, saying the EU has "not treated the United States fairly on trading".
He also said Mrs May had not listened to his advice on how to do a Brexit deal, saying: "I would have done it much differently.
"I actually told Theresa May how to do it but she didn't agree, she didn't listen to me. She wanted to go a different route," he said.
Tom Newton Dunn, the Sun journalist who interviewed Mr Trump, said the US president seemed "sensitive" and knew about the "Trump baby" inflatable.
"He's really quite stung by the criticism he's been getting," said Mr Newton Dunn. "He knew all about the baby blimp. I think it hurt him."
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.
But Foreign Office minister Sir Alan Duncan said things had "moved on" since Mr Trump's interview - which was carried out before he arrived in the UK - and the mood at Thursday night's dinner was "fantastically positive and did focus a lot on trade".
The government does not see Mr Trump's behaviour as "rude", said Sir Alan, adding: "Donald Trump is a controversialist. That's his style."
Mayor of London Sadiq Khan 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."
And, responding to Mr Trump's criticism of his response to terrorism, Mr Khan said it was "interesting" that he "is not criticising the mayors of other cities" which have also experienced terror attacks.
Meanwhile, Shadow Foreign Secretary Emily Thornberry said the PM "should be standing up to [Mr Trump]" after he "slagged her off", instead of holding his hand.
Mr Trump briefly held Mrs May's hand as they went up the stairs
Mr Trump's comments came on the same day the UK government published its proposal for its long-term relationship with the EU.
The plan is aimed at ensuring trade co-operation, with no hard border for Northern Ireland, and global trade deals for the UK. Mrs May said the plan "absolutely delivers on the Brexit we voted for".
But after ministers reached an agreement on the plan at Chequers a week ago, leading Brexiteers Boris Johnson and David Davis resigned from the cabinet.
Mrs May and Mr Trump are watching 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 also being attended by 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.
