Trump said in an interview published on Friday that May had not followed his advice on Brexit - one of a number of comments that were seen as a damning critique of her exit plans and added fuel to the raging debate over May's plans to leave the EU.
Revealing the surprising suggestion on the BBC's Andrew Marr program on Sunday, May appeared to stifle what appeared to be a chuckle of disbelief at Trump's outlandish idea.
"Once the Brexit process is concluded and perhaps the United Kingdom has left the European Union, I don't know what they're going to do but whatever you do is OK with me, that's your decision", he said.
"Don't walk away from negotiations, because then you'll be stuck".
Trump called May "a very tough, very smart, very capable person", adding that he would "much rather have her as my friend than my enemy".
"So let's just keep our eyes on the prize here".
Before the paper was published, Brexit Secretary David Davis and Foreign Secretary Boris Johnson resigned, saying it would not deliver the Brexit people voted for in the 2016 referendum. The deal she is pursuing "is not what the people voted on" and will affect trade with the United States "in a negative way", he said.
"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, but it is not an easy thing".
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On Thursday, AT&T said: "The court's decision could hardly have been more thorough, fact-based and well researched". CNN has always been a target for Trump , calling the Time Warner-owned news network "fake news".
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The New York Times ran a website headline saying: "With May's Government Teetering, Trump Gives It a Shove". The i reports that 100,000 people took to the streets of London to protest against the visit of Mr Trump .
However, she insisted that she had yet to see a "workable alternative" to the proposals - agreed by her Cabinet at Chequers - that would ensure trade remained as "frictionless" as possible while avoiding the return of a hard border in Northern Ireland.
In her Mail on Sunday letter, May warned parliamentarians seeking to scuttle the plan - and also those trying to force amendments to strengthen post-Brexit EU ties - that they risk causing "a damaging and disorderly Brexit".
Both could speak in the debate, now due to start at 1430 GMT and end with votes at 2100, and they may have an important influence over how many colleagues are willing to speak out. I wouldn't say advice.
"It will certainly be something that we bring up and talk about", Mr Trump said.
"I'm sure Theresa May does not want to split the Conservative Party and therefore she will find that the inevitable outcome of the parliamentary arithmetic is that she will need to change it (the Brexit policy) to keep the party united", Rees-Mogg said.
"We're going to be able to cut tariffs, we're going to be able to change quotas, we're going to be able to have freedom on services, we're going to be able to have bilateral investment deals", May said on the BBC.
She acknowledged some MPs had concerns about her plan for a "common rule book" with the EU for goods and customs traded within what she called a new "UK-EU free trade area".