Boris Johnson hailed the comments as he arrived at a meeting of EU foreign ministers in Brussels on Monday.
Trump said Britain leaving the EU would "end up as a great thing" and promised to work for a trade deal with post-Brexit Britain "quickly and done properly", speaking in an interview with The Times newspaper.
Britain's foreign minister Boris Johnson hailed the comments as he arrived at a meeting of EU foreign ministers in Brussels on Monday.
"I think it is very good news that the United States wants to do a good free trade deal with us and wants to do it very fast," he said.
Trump's comments did little to reassure investors, however, following reports in British media over the weekend that May is planning to announce a hard line on Brexit in a major speech on Tuesday.
Sterling plunged to $1.1986, its lowest level since October's "flash crash" that had sent it to a 31-year low of $1.1841, in morning trading.
"The market is now positioning for some fairly punchy rhetoric from Theresa May," said Chris Weston, chief market strategist at IG, an online trading company.
"This idea of 'hard Brexit' and a clean break from the single market seems increasingly likely, with the government making a bid to gain full control over immigration," Weston said.
Sterling "is an out-and-out political currency... and the prospect of volatility here is now very high" despite some positive macroeconomic data, he said.
Kathleen Brooks, research director at City Index, said: "It's all about market dynamics, and right now the balance of market participants are shorting the pound.
"It's a bit like a tipping point, once a trend gets critical mass, like the GBP downtrend, then news headlines can have big impacts, as they generate another wave of selling," she said.
May has said she wants to cut down on EU immigration while retaining the "best possible" access for British companies to Europe's single market.
EU leaders have said these are incompatible and market access would only be granted in exchange for accepting the EU's rules on free movement of people.
In an interview with Germany's Welt am Sonntag newspaper over the weekend, finance minister Philip Hammond hinted at the new "hard Brexit" line.
Answering a question about whether Britain could become a tax haven, he said it would be forced to "change our economic model" to remain competitive if it is shut out from access to the single market.
May aims to launch two years of EU departure negotiations when she triggers the Article 50 exit process by the end of March, although a legal challenge is still pending before the country's Supreme Court.
She has been under pressure to reveal her proposals for the talks that will establish the future relationship between Britain and the EU.
Matching reports in several weekly newspapers said she was prepared to accept pulling out of the single market, the European customs union and the European Court of Justice, in order to regain control of EU immigration.
"She's gone for the full works. People will known when she said Brexit means Brexit, she really meant it," a government source was quoted as saying in The Sunday Telegraph.