The US leader and his wife Melania were greeted by the Queen on Friday afternoon following talks with Theresa May at Chequers.
It followed criticism of The Sun newspaper as "fake news" by Mr Trump after it published an interview in which he appeared to criticise Mrs May's handling of Brexit.
The US President told reporters at Chequers he thought the Prime Minister was doing a "terrific job", adding that he would be happy with however Brexit negotiations conclude.
Watch in full: Donald Trump meets the Queen amid huge protests
Mr Trump insisted that he had not criticised Mrs May over Britain's departure from the EU, expanding on comments made earlier on Friday that the pair had "probably never developed a better relationship".
Both leaders also told a press conference about their hopes of striking a UK-US trade deal with no bars post-Brexit.
Mr Trump appeared to row back on the comments whilst talking to reporters, claiming The Sun's story missed out on positive remarks he made about Mrs May.
Claiming that the White House records all interviews, Mr Trump said: "It's called fake news and we solve a lot of problems with the good old recording instrument."
Speaking positively about the prospects of striking a US-UK trade deal, Mrs May said: "There will be no limit to the possibility of us doing trade deals around the rest of the world once we leave the European Union on the basis of the agreement that was made here at Chequers and that I've put forward to the European Union."
And on the same topic, Mr Trump said: "The only thing I ask of Theresa is that we make sure we can trade and we don't have any restrictions because we want to trade with the UK and the UK wants to trade with us.
"We are by far their biggest trading partner and we have just a tremendous opportunity to double, triple, quadruple that."
Earlier on Friday, Mr Trump said his relationship with Mrs May was "very strong", and later added that the UK-US bond was the "highest level of special".
Nevertheless, Mr Trump's interview with The Sun will likely have strained talks, with Mrs May already under significant pressure from Brexiteers.
In the interview, Mr Trump voiced concern that Mrs May's Brexit plans would leave Britain too closely aligned with the European Union.
He said he would have done the Brexit negotiations "much differently" and claimed the Prime Minister did not listen to his advice.
Speaking to reporters, he denied giving Mrs May advice, saying: "I did give her a suggestion. I wouldn't say advice.
"I think she found it maybe too brutal because I could see that - I don't know if you remember what I said. I did give her a certain amount... I gave her a suggestion, not advice, I wouldn't want to give her advice, I'd give her a suggestion.
"I can fully understand why she thought it was a little bit tough and maybe someday she will do that - if they don't make the right deal she might very well do what I suggested that she might want to do."
Mrs May said: "Lots of people give me advice about dealing with the European Union. My job is actually getting out there and doing it."
Meanwhile, thousands of protesters have taken to the streets in London and other cities to protest the presence of Mr Trump.
A Trump baby blimp was raised up in the air as part of demonstrations ahead of rallies.
Cheers greeted the 20ft high inflatable caricature's take-off in central London at around 9.30am.
Mr Trump will not personally witness the blimp's flight, which received support from his nemesis, London Mayor Sadiq Khan, but the Republican said it had made him "feel unwelcome" in London.
Labour leader Jeremy Corbyn and Labour MP David Lammy both addressed crowds at Trafalgar Square as the protests peaked in the late afternoon.
In the controversial interview, released while Mr Trump and First Lady Melania were being entertained by the Prime Minister at Blenheim Palace, the president said: "If they do a deal like that, we would be dealing with the European Union instead of dealing with the UK, so it will probably kill the deal.
"If they do that, then their trade deal with the US will probably not be made."
The comments, following on from the morning press conference, will be a cause of great concern for Mrs May.
She had used the Blenheim black tie dinner with political and business leaders to press Mr Trump on the benefits of a free trade deal after Brexit.
Addressing the 100-strong group the Prime Minister said there was an “unprecedented” opportunity to do a deal that boosted jobs and growth in both countries.
But Mr Trump appeared to link Brexit to the current trade dispute between the US and EU over steel and aluminium.
He told The Sun Mrs May’s plan would affect trade "unfortunately in a negative way". He added: "We have enough difficulty with the European Union.
"We are cracking down right now on the European Union because they have not treated the United States fairly on trading.
"No, if they do that I would say that that would probably end a major trade relationship with the United States."
The Trumps landed in Air Force One at Stansted Airport at lunchtime on Thursday to kick-start a four-day working visit to Britain as protests against his trip began.
Speaking to reporters in Belgium after a fiery Nato Summit, Mr Trump had described the UK as a “hot spot right now with a lot of resignations” and dismissed the Prime Minister’s Chequers plan on the next stage of Brexit.
"I would say Brexit is Brexit,” he told reporters in Belgium.
"The people voted to break it up so I would imagine that’s what they would do, but maybe they’re taking a different route, I don’t know if that is what they voted for."
He added that it seemed as if the UK was “getting at least partially involved back with the European Union”.
"I’d like to see them be able to work it out so it could go quickly, he said.
It comes just days after Mr Trump declined to say whether Mrs May should remain in post, said he had “always liked” Boris Johnson, who quit as foreign secretary over the Chequers agreement, and described the UK as being in “turmoil”.
He reiterated his praise for the former foreign secretary on Friday, telling reporters: "He's been very nice to me. He's been saying very good things about me as president.
"I think he thinks I'm doing a great job. I am doing a great job, I can tell you, just in case you haven't noticed.
"Boris Johnson, I think, would be a great prime minister."