Trump said in an interview published in Rupert Murdoch's Sun newspaper on Friday that May had not taken his advice on Brexit, which was just one of many comments made by the president about her exit plans.
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.
She quickly followed with a laugh: "Actually we're going into negotiations with them".
At the press conference, however, Trump called it "fake news", saying that the report omitted the positive things he said about May.
"What the president also said at that press conference was don't walk away".
The 20ft inflatable depicting the USA president as a nappy-clad baby holding a mobile phone has been brought north of the border after being flown above Parliament Square in London on Friday.
Trump, however, later denied ever criticizing May, claiming that The Sun failed to include the several compliments he showered May with during his interview.
But our Brexit deal for Britain achieves exactly this - and it can work.More news: President Trump Reportedly Threatened to Pull the U.S. Out of North Atlantic Treaty Organisation
A White Paper published on Thursday fleshed out details of her plan, which advocates close links with the European Union on trade in goods, but not services.
Mr Baker said the public would eventually "know they've been betrayed", adding: 'If we spoil and wreck Brexit, which Chequers does, then we will get Jeremy Corbyn.
The Daily Telegraph reported more than 100 MPs had joined the group - more than double the 48 needed to submit letters of no confidence in the Prime Minister to force a leadership contest.
While May was dismissive of Trump's suggestion to launch legal proceedings against Brussels, she did highlight his concerns over "no deal" with Brussels.
Her Conservative government is deeply split between supporters of a clean break with the European Union and those who want to keep close ties with the bloc, Britain's biggest trading partner.
"I recommended her something, I gave her an option, I'd rather not tell you what that option is, but I think she might", he told The Mail on Sunday newspaper. As President Trump has said, I'm a tough negotiator. Steve Baker, the Brexit minister who quit along with Davis, accused her in The Sunday Telegraph of presiding over a "cloak and dagger" plot to undermine Brexit.
David Davis, who resigned as Brexit Secretary last week, is expected to set out his objections to her strategy in a speech to the Commons today. "And I think it probably would have worked", Trump said. He was reported saying in the British newspaper The Mail on Sunday, "She is an incredible woman, she is so sharp, she is so lovely, when I say handsome - inside and out".
For the common rule book only covers industrial goods and agricultural products and only those rules which are necessary to ensure free flow at the border.