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President Trump and Prime Minister Theresa May of Britain at the White House in January. She was the first foreign leader to make an official visit after Mr. Trump’s inauguration. Credit Stephen Crowley/The New York Times

LONDON — It began with smiles and holding hands in the White House. Then she told him he was wrong to send a bunch of tweets the way he did. So he told her to mind her own business.

That, roughly, has been the trajectory of the ties — which some Britons call a special relationship — between President Trump and Theresa May, the British prime minister who hurried to Washington to be the first foreign leader to pay an official visit to the new American leader after his inauguration.

And now the relationship may be put to the test once more. The United States ambassador to Britain, Woody Johnson, said on Tuesday that he believed that Mr. Trump would come to London to dedicate the new United States Embassy, reviving a debate among Britons about what kind of welcome Mr. Trump might expect.

The presidential trip has been an on-off kind of affair — more off than on, it has sometimes seemed — ever since Mrs. May invited Mr. Trump on behalf of Queen Elizabeth II to pay an official visit.

But its importance has been magnified by Britain’s plans to leave the European Union in 2019, forcing the country to form new trade relationships with major economic powers to offset the cost of withdrawing from the 28-nation European bloc.

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Mr. Trump’s personality has injected an unpredictable element into the long alliance with Britons, whose perception of American leaders has sometimes been ambiguous.

Former Prime Minister Tony Blair, for instance, was castigated frequently for being so close to President George W. Bush in the run-up to the invasion of Iraq in 2003 that he was mocked as America’s “poodle.”

For all that, Mr. Trump’s break with the traditional orthodoxies of diplomacy has taken the perception that there is a trans-Atlantic disconnect to higher levels.

Mr. Johnson, the American ambassador, told the BBC on Tuesday that Mr. Trump was “never going to go down the path of a lot of politicians” by behaving in a “namby-pamby” way. “Maybe he’ll ruffle feathers,” he said of the president. “There’s no question that maybe some feathers were ruffled.”

That was a reference to the most recent spat, when Mr. Trump retweeted three posts that had originated with a small, extreme right-wing group called Britain First. The posts included video clips portraying Muslims in a hostile way that were widely interpreted as Islamophobic.

Displaying an uncharacteristic bluntness in dealings with an American president, Mrs. May said Mr. Trump had been “wrong” to send the messages. The president responded on Twitter: “Don’t focus on me, focus on the destructive Radical Islamic Terrorism that is taking place within the United Kingdom. We are doing just fine!”

That message struck a raw nerve, because Britain has been the target of five terror attacks this year in which dozens of people have died, and the security services also claim to have thwarted several more, including a conspiracy to murder Mrs. May herself.

During one attack in July, Mr. Trump also seemed to take a jab at Sadiq Khan, the first Muslim to be elected mayor of London. “At least seven dead and 48 wounded in terror attack and Mayor of London says there is ‘no reason to be alarmed,’ ” Mr. Trump said on Twitter on June 4, after an attack on London Bridge and a nearby market.

Mr. Khan has been no less sharp-tongued about Mr. Trump, saying the American president should not visit Britain because “his policies go against everything we stand for.”

Such exchanges have fanned an undisguised hostility among Mr. Trump’s critics in Britain, where there have been calls for the president’s visit to be called off or met with protests. Indeed, Mr. Trump’s reposting of the extreme-right video clips brought a rare unity among British politicians in condemning him.

Against that backdrop, British diplomats have emphasized that in areas such as security and intelligence, the two countries have deep ties whose durability and importance transcend individual leaders.

“What we have here is not just Trump the man but the institution” of the American presidency, said Sir Christopher Meyer, a former British ambassador to Washington.

The BBC quoted unidentified sources on Tuesday as saying a working visit was possible in late February to dedicate the new American Embassy, which has been relocated from Grosvenor Square in central London to the Nine Elms area south of the River Thames.

Speaking to the BBC, Mr. Johnson, the American ambassador, said no date had been set for the president to visit Britain, but “absolutely, I think he will come.”

“It hasn’t been officially announced, but I hope he does,” the ambassador said.

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