When asked whether Mrs. May should stay in her job, Mr. Trump adopted the pose of a disinterested observer, pronouncing this a question for “the people.” He then suggested that speaking with President Vladimir V. Putin of Russia, which he is scheduled to do next week, might be easier than conversing with Mrs. May, who will host a dinner and a lunch with the president during his stay.

Yet the prime minister has little choice but to swallow her pride and get on with it, as she herself likes to say, because she wants Mr. Trump’s help for a trade deal with the United States after Britain’s withdrawal from the European Union, known as Brexit. And, anyway, compared with wrestling with her government, which this week suffered three ministerial resignations, spending a couple of awkward days with Mr. Trump might actually be a relief.

Beyond the psychodramas and the expected protests at every stop on Mr. Trump’s itinerary, the visit also raises more profound questions about the durability of the special ties between London and Washington at a time when Mr. Trump is attacking the basic institutions of the postwar international order and with Britain on the verge of leaving the European Union.

No one argues that the relationship between Mr. Trump and Mrs. May is anything like that between such venerated predecessors as Winston Churchill and Franklin D. Roosevelt, or akin to the political romance of Ronald Reagan and Margaret Thatcher. But is it a sign of bad personal chemistry or of something deeper, a widening divide that might never be sewn back up?

Optimists in Britain say the deep economic, military, intelligence-sharing and cultural ties across the Atlantic are strong enough to survive the occasional Trump Twitter storm.