He once said his chances of becoming prime minister were “as good as the chances of finding Elvis on Mars,” but rumpled Boris Johnson, a former mayor of London, a former foreign secretary and an unapologetic supporter of Britain’s exit from the European Union, is now the odds-on favorite to succeed Theresa May when the votes of Conservative Party members are counted next week.

Mr. Johnson is considered such a strong favorite over Jeremy Hunt, the current foreign secretary, that the speculation has moved on to what the change of leadership in London will mean for the trans-Atlantic relationship and whether the mutual admiration society between Mr. Johnson and President Trump will bring major changes to the now-rocky U.S.-British relationship.

As the next prime minister, Mr. Johnson argues, he would usher in a new era of camaraderie in bilateral relations with Washington.

“I’ve got a good relationship with the White House, and I have no embarrassment in saying that I think it’s very important that we have a strong relationship with our most important ally,” Mr. Johnson told voters at a campaign rally Thursday night.

Mr. Trump, who has shown no reluctance to weigh in on domestic British politics, has made clear what he thinks of the candidate who shares his views on Brexit and his skepticism of the European Union.

“I think Boris would do a very good job. I think he would be excellent …,” Mr. Trump told the British newspaper The Sun in June, just as Mrs. May was preparing to formally step down. “He has been very positive about me and our country.”

Nile Gardiner, director of The Heritage Foundation’s Margaret Thatcher Center for Freedom, said in an interview that he expected a “strong and robust partnership” between Mr. Johnson and Mr. Trump.

“They’re both very charismatic figures, they’re both leaders who have broken the political mold,” he said. “They’re both figures who challenge conventional wisdom and are big-picture politicians.”

Freddy Gray, deputy editor of the conservative British weekly The Spectator, said that after Brexit, “the potential of a Trump-Boris alliance is arguably Britain’s biggest hope.”

“The two men have a chemistry that goes beyond their unusual hairstyles and appreciation of younger women,” Mr. Gray wrote in an analysis last week. “Both grasp that a profound shift is taking place in politics, one that has propelled people like them to power. They also sense, in the way that macho beasts often do, a certain destiny in each other.

“Trump likes Britain, Brexit and Boris; it’s that simple,” Mr. Gray added.

Focused on Brexit

Mr. Johnson’s top priority will be Brexit. He was one of the most prominent “leave” voices in the shocking 2016 national referendum and has said he is prepared to take Britain out of the European Union whether or not an exit deal with Brussels is reached by the Oct. 31 deadline. Mr. Trump has repeatedly dangled the prospect of a bilateral trade deal with the U.K. once it leaves the bloc.

“The more determined we are to pursue ‘no deal,’ the less likely we will have to deploy it,” Mr. Johnson told Parliament recently. “I do not want it. But to have an orderly exit from the EU, it is vital you prepare.”

Mr. Trump appears to agree. He resorted to Twitter again last week to criticize how Mrs. May handled the Brexit talks and to fault her for failing to follow his advice.

“What a mess she and her representatives have created,” Mr. Trump tweeted. “I told her how it should be done but she decided to go another way.”

An explosive diplomatic furor in the middle of the Johnson-Hunt leadership fight has Mr. Johnson’s critics in Britain claiming he will be too friendly to the U.S. and too deferential to Mr. Trump in the vaunted “special relationship.”

Mr. Trump responded angrily after the leak last week of a series of cuttingly critical cables on his temperament and his administration by British Ambassador Kim Darroch. Mrs. May and Mr. Hunt condemned the leaks but defended the career diplomat, saying he was doing his job in giving his candid assessment of the country where he was posted.

“Who chooses our ambassadors is a matter for the United Kingdom government and the United Kingdom prime minister, and I have made it clear if I am our next prime minister, the ambassador in Washington stays because it is our decision,” Mr. Hunt said in a candidates debate last week in London.

Mr. Johnson was far more equivocal. He declined to say whether he would allow the ambassador to stay on if he is elected and said the U.S. relationship is of “fantastic importance” and that Mr. Trump “had been dragged into a British political debate.”

Mr. Darroch made the matter moot by resigning his post Thursday, but Mr. Johnson’s stand sparked controversy in Britain. Mr. Johnson said his debate comments had been “misrepresented” to Mr. Darroch, leading him to resign, but critics said the incident could be a sign of things to come.

“Johnson’s kowtow to Trump before he has even entered Downing Street is likely to be the first of many acts of abasement as Britain wakes up from the dream of Brexit to the bleak reality ahead,” Denis Staunton, London editor of the Irish Times, wrote in an opinion piece last week.

Differences on Iran

Not all may go smoothly between Mr. Trump and Mr. Johnson, however.

While Mr. Trump has slammed the 2015 Iran nuclear deal and last year withdrew the U.S. from the international pact, Britain under Mrs. May has joined other European powers in an effort to preserve the deal and to find a way for Iran to get around harsh new U.S. economic sanctions.

Mr. Johnson told Reuters news service last week that although he saw “disruptive behavior” from Iran, he remained an advocate for the deal.

“I continue to believe that engaging with Iran and persuading Iran not to go forward with a nuclear weapons program is the right way forward for our country and for the region,” he said.

Mr. Gardiner predicted that a Prime Minister Johnson would start moving Britain away from Mrs. May’s staunch support of the Iran deal.

“I expect with a new British government, you’re going to see a different approach taken towards the Iran deal. I think that Britain will increasingly move away from EU policy positions,” he told The Washington Times.

He said Ms. May is “very attached to the Iran deal,” an EU initiative and a foreign policy that he doesn’t expect to align with the next U.K. government.

Mr. Trump and Mr. Johnson also share a skepticism of immigration and open borders. The U.K., Mr. Johnson has said, must be “tougher on those who abuse our hospitality,” and a major thrust of the pro-Brexit campaign in 2016 was support for British reclamation from EU control of its immigration and labor policies.

Sounding much like Mr. Trump, Mr. Johnson has said of immigration, “We must be much more open to high-skilled immigration, such as scientists, but we must also assure the public that as we leave the EU we have control over the number of unskilled immigrants coming into the country.”

Mr. Gardiner said the bilateral relationship could flourish under Mr. Johnson because he is an unabashedly “pro-American politician” at a time when that sentiment is not widely shared among many in the top levels of British government.

“I think that Boris Johnson as prime minister will result in a reinvigorated U.S.-U.K. relationship,” Mr. Gardiner said. “Johnson has already established a good rapport with Donald Trump. It’s expected he’ll have a far warmer relationship than Theresa May did.”

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