May has given her full support to Washington envoy Kim Darroch after he described Trump's administration as "dysfunctional" and "diplomatically clumsy and inept" in memos that were leaked to a British newspaper on Sunday.
While ministers and May's spokesman have stated the leaked cables were only a partial reflection of Darroch's assessments and did not reflect the government's views, they have infuriated Trump who fired off a second series of scathing tweets.
"The wacky Ambassador that the UK foisted upon the United States is not someone we are thrilled with, a very stupid guy," Trump wrote, describing Darroch as a "pompous fool".
"He should speak to his country, and Prime Minister May, about their failed Brexit negotiation, and not be upset with my criticism of how badly it was handled. I told @theresa_may how to do that deal, but she went her own foolish way - was unable to get it done. A disaster!"
May is due to leave office this month and has previously clashed with Trump over issues from Muslim relations to the 2015 Iran nuclear deal.
However, the timing of the discord comes as Britain is hoping to strike a major trade deal with its closest ally after it leaves the European Union, an exit scheduled for Oct. 31.
LEADERSHIP RACE
The two contenders to replace May, former London mayor Boris Johnson and foreign minister Jeremy Hunt, have both indicated they could support leaving the EU without a deal, making a future agreement with the United States even more important.
Johnson, the favourite in the race, hinted that he shared Trump's view of May's handling of Brexit and said the United States would remain Britain's top military and political partner.
"I've got a good relationship with the White House and have no embarrassment in saying that", he told BBC TV. "I myself have said some pretty critical things about the Brexit negotiations so far."
But his rival, Hunt, challenged Trump, saying his comments were "disrespectful and wrong" to May and the country.
"You said the UK/US alliance was the greatest in history and I agree," he said on Twitter. "But allies need to treat each other with respect as @theresa_may has always done with you. Ambassadors are appointed by the UK government and if I become PM our Ambassador stays."
Britain's trade minister Liam Fox said he would apologise to Trump's daughter Ivanka whom he was due to meet during a visit to Washington this week. Darroch had been due to attend that meeting but has now pulled out, the BBC reported.
On Monday, Trump had tweeted that he would no longer deal with the diplomat and that it was "good news" that Britain would soon have a new prime minister.
Before Trump's latest broadside, May's spokesman said Darroch would stay in place with the government's full support and that envoys had to feel confident they could share their frank assessments.
"She (May) said we had made clear to the US how unfortunate this leak is, and that the extracts were selective and do not reflect the closeness of our relationship," the spokesman said.
"SERIOUS CONSEQUENCES" FOR LEAKER
In his confidential memos dating from 2017 to the present, Darroch had said reports of in-fighting in the White House were "mostly true" and last month described confusion within the administration over Trump's decision to call off a military strike on Iran.
"We don't really believe this Administration is going to become substantially more normal; less dysfunctional; less unpredictable; less faction riven; less diplomatically clumsy and inept," Darroch wrote in one cable.
British officials have launched an inquiry to find out who was responsible for the leak.
Christopher Meyer, a former British ambassador to Washington, said there was a "possible range of villains".
"It was clearly somebody who set out deliberately to sabotage Sir Kim's ambassadorship, to make his position untenable and to have him replaced by somebody more congenial to the leaker," he told BBC radio.