It is unclear how such a lawsuit would work for Britain, a member of the European Union, but Trump has often threatened lawsuits in dealmaking.
The two leaders have disagreed on how May should handle the exit from the bloc, with Trump frequently haranguing her to hurry the process. Trump has often begun phone calls with her by asking her to rush the process.
Trump also continued his public criticisms of the E.U., calling it a “foe” in a CBS interview that aired Sunday.
“I think we have a lot of foes. I think the European Union is a foe, what they do to us in trade,” Trump said in the interview. “Now, you wouldn’t think of the European Union, but they’re a foe. Russia is a foe in certain respects. China is a foe economically; certainly they are a foe. But that doesn’t mean they are bad.”
Saying the E.U. is “very difficult,” Trump berated the 28-member union over trade — complaining that European nations have taken advantage of the United States. He also continued criticisms of NATO that he began at the start of his Europe trip, saying other nations in the alliance weren’t spending enough money on national defense.
“Don’t forget, both my parents were born in E.U. sectors, okay?” Trump said in the CBS interview. “You know I love those countries. I respect the leaders of those countries. But in a trade sense, they’ve really taken advantage of us, and many of those countries are in NATO and they weren’t paying their bills.”
European Council President Donald Tusk quickly fired back on Twitter, saying, “America and the EU are best friends. Whoever says we are foes is spreading fake news.”
After he landed in Europe last week, Trump conducted an interview with the Sun, a British tabloid, in which he criticized May.
“I would have done it much differently,” he told the Sun. “I actually told Theresa May how to do it, but she didn’t listen to me.”
He added: “The deal she is striking is a much different deal than the one people voted on.”
He also described his advice as “brutal” but did not say what the advice was. May is politically vulnerable because of Brexit, analysts say, and Trump’s comments drove nonstop headlines questioning her policy.
His comments to the Sun led to a furor in London, and he eventually seemed to backtrack, saying he would support May no matter what she did.
“Interestingly, what the president also said at that news conference was ‘Don’t walk away,’ ” May told the BBC.
“ ‘Don’t walk away from those negotiations because then you’ll be stuck.’ So I want us to be able to sit down to negotiate the best deal for Britain,” she added.
May has taken a more moderate stance, releasing a lengthy white paper last week on exiting the union in spring 2019. That has led to some tension within the British government, with Foreign Minister Boris Johnson resigning and saying her plan is not what people voted for.
Trump also caused headaches by endorsing Johnson, a foe of May, as a great potential prime minister.
Kim reported from Helsinki.