Mr. Trump, who loves tough guys, brought Churchill front and center after assuming the presidency. Last year, he restored to the Oval Office a bust of the prime minister that had been moved during the Obama administration.

And the grimace Mr. Trump has used in official portraits and on the world stage? That, too, was born of an effort to look like Churchill.

The real Churchill, sometimes to his chagrin, was painted to look much like a joyless bulldog, with squinted eyes and a protruding lower lip.

Last winter, as Mr. Trump was adjusting to his presidency, he was host to a screening of “Darkest Hour” at the White House. The film, which lionizes Churchill’s extraordinary struggle to see Britain through a devastating war and face down one of history’s most destructive monsters, struck a chord with the president. Mr. Trump has said he usually has trouble sitting through movies, and will get up and leave if he is uninterested.

“Darkest Hour” stuck with the president as he made plans to visit Britain. In a phone call last winter, he told Mrs. May that she could become this generation’s Churchill, a British official said on Thursday. The official, who wasn’t authorized to discuss the conversation and spoke on the condition of anonymity, was confirming a detail first reported by The Telegraph.

An official involved in the planning for the visit said there was even discussion of having the president visit the underground London bunker used by Mr. Churchill as a war room during World War II. Those plans were eventually cut out of the final agenda.

Mr. Trump is not the only Republican who admires Churchill. John Boehner, the former House speaker, and his successor, Paul D. Ryan, have held parties honoring Churchill at the Capitol. Other conservative figures like Jerry Falwell Jr., Michael Savage and Mike Huckabee have compared Mr. Trump to Churchill.