On Friday, Trump said that at a gala dinner at Blenheim Palace the night before he and May spoke for nearly 90 minutes and had "probably never developed a better relationship".
The president also hit out at immigration into Europe, linking it with crime.
"I would say I would give our relationship in terms of grade the highest level of special", Mr Trump said.
In the meantime, the United Kingdom wants to remain part of the European Arrest Warrant (EAW) and is also seeking full participation in Joint Intelligence Teams (part of Eurojust), also discussing future migration relations.
Asked earlier in the day whether he regrets his interview with British tabloid The Sun, Trump very visibly shook his head and rolled his eyes, turning to his aides with a look that showed his displeasure with the question.
"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.
Downing Street had hinted in advance they were expecting a more emollient tone from the president. A week that started with a rare show of unity on Brexit by her government then led to two senior ministers quitting within 24 hours of each other and rebels within her own party dismissing her plan as a betrayal of what voters decided in the 2016 referendum.
The President revealed that he had "told" Theresa May how to do a Brexit deal (he does claim that dealmaking comes easy to him) but that the Prime Minister, for reasons best known to her, had ignored her advice. "I would have done it much differently", he added.
In his interview with The Sun, Trump slammed May over her "soft" Brexit plan, which has put May's standing as the head of the Conservative Party in jeopardy. "She wanted to go a different route", he told the paper.
The lack of divergence on rules in the farming sector is one of the main concerns of Tory MPs opposed to May's plan, and they will see Trump's statement as proof that the Brexit plan will indeed thwart a trade deal.
"This will be a hard negotiation; and I don't necessarily think that because the British position has become more credible that that's necessarily the final picture".
White House spokeswoman Sarah Sanders later sought to limit the damage, saying Trump was "thankful for the wonderful welcome" he received in Britain.
"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.
Mr Varadkar said: "what we are not preparing for is a border between north and south but we will need to make preparations for changes at our ports and airports".
No 10 sent the Foreign Office minister Sir Alan Duncan out to bat on the airwaves, where he insisted the interview had not been damaging and the visit would "most definitely" be a success. "I think you are losing your culture".
Before heading for Scotland, the president and Queen Elizabeth II became acquainted over tea at Windsor Castle.
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