LONDON — President Trump hasn’t even confirmed the exact dates for his state visit, but British politicians are already fretting — if not in an outright panic — over how to cope with the politically dicey drop-by, not least because of the protests that are likely to erupt once he arrives.

Enter British Prime Minister Theresa May, who has come up with a plan: She wants to meet the president in the remote Scottish Highlands, where she could not only welcome the leader of the free world with pomp and circumstance worthy of the vaunted “special relationship,” but also keep Mr. Trump away from outbursts that might jeopardize a trade deal she desperately needs because of the U.K.’s divorce from the European Union in two years.

Whether the sizable anti-Trump contingent in England and Scotland will let her thread that needle is another question.

“Theresa May is obviously worried about the kind of reaction President Trump will get in London,” said David Torrance, a Scottish political pundit and author. “Sending him up north to Scotland looks like a great move on her part to keep the Brexit diplomacy running smoothly — until you realize that the president is just as controversial north of the border.”

When she visited the White House in February, Ms. May extended the invitation and promised Mr. Trump a lavish state visit with Queen Elizabeth.

Mr. Trump was an early enthusiast for Brexit, and his stated preference for one-on-one trade deals over multilateral pacts such as the North American Free Trade Agreement or a proposed deal with the European Union was music to the ears of Britons eager to break free from Brussels’ grasp.

With Brexit scheduled for April 2019, Ms. May and others are eager to strike a trade deal with the U.S. before the breakaway takes effect. Inviting Mr. Trump to Britain and promising him a state visit were intended to push along the trade negotiations.

British Foreign Secretary Boris Johnson said Mr. Trump assured him during a White House visit in January that Britain was “first in line to do a great free trade deal with the United States.”

Mr. Trump’s visit, originally slated for June, has been moved to the fall in the expectation of demonstrations — which look to be huge.

“They don’t want what will be one of his first big foreign trips to be overshadowed,” said a British official, citing State Department handlers wary of Londoners staging television camera-ready protests against Mr. Trump, according to documents that the British press obtained last month.

The British harbor mixed opinions of the U.S. president. After Mr. Trump’s election victory, protesters took to the streets across the British Isles. In February, Parliament debated banning Mr. Trump from Britain after he issued an order temporarily barring travelers from seven majority-Muslim countries. Parliament opted not to ban the president even as a U.S. federal court was blocking Mr. Trump’s travel order.

The flamboyant, outspoken billionaire real estate developer appears to rub many buttoned-down, stiff-upper-lip Britons the wrong way with his comments regarded as sexist and his penchant for conspicuous consumption.

The White House over the weekend denied an anonymously sourced London Times story that said Mr. Trump was demanding a ceremonial ride through London in one of Queen Elizabeth’s gold-laced royal carriages as an “essential element” of his October state visit, the same mode of transportation used by world leaders such as Russian President Vladimir Putin on their visits. A White House spokeswoman called the account “completely false” and said planning for the trip had not even begun.

Trump’s fans

Still, the American president has his supporters. Britain’s most outspoken advocate for quitting the EU during the campaign, U.K. Independence Party leader Nigel Farage, is a staunch admirer of Mr. Trump. The two men have bonded over their surprise victories at polls that contradicted political pundits’ forecasts, and Mr. Farage even campaigned with the Republican nominee last year.

Mr. Trump’s outspoken skepticism of the EU and his rejection of unchecked immigration find a strong echo in the rural and industrial heartlands of Britain as well.

Official anxiety on both sides of the Atlantic is a British “storm in a teacup,” said Stuart MacLennan, a Coventry University political scientist who supports Brexit.

“By all means, protest his views. I personally share very few, if any of them,” said Mr. MacLennan. “But protesting his presence makes no sense. We invite far less pleasant leaders to the U.K. on a regular basis, and hardly anyone bats an eyelid. I’ve always taken the view that Britain is better off working with the world.”

Queen Elizabeth has received several important but controversial historical figures, including Japanese Emperor Hirohito, who reigned during World War II; Chinese President Hu Jintao, who is thought to have played a role in the brutal suppression of activists in Tiananmen Square in 1989; and Mr. Putin.

British officials are debating where in Scotland to hold the meetings. The most likely venue is the queen’s Balmoral Castle in Aberdeenshire, where high security is the norm whenever the monarch is in residence.

The castle is also a short drive from Mr. Trump’s luxury golf resort at Menie near the city of Aberdeen. Mr. Trump visited the site many times before he won the presidency and has called it “the greatest golf course in the world.”

Scotland might be more remote than London, but its voters are more left-wing and pro-Europe than the English. Rory Scothorne, 24, an author and co-organizer of the Scotland Against Trump coalition, said the president can expect a true Highland welcome if he touches down, come rain or shine.

“If the Conservative Party thinks they can hide their dealings with Trump away from the public eye, they won’t find a safe haven in Scotland,” Mr. Scothorne said. “Wherever he shows up, we’ll show up too.”

By the time Mr. Trump touches down in Albion in Air Force One, his predecessor already will have made headlines in Scotland. Former President Barack Obama, who is popular among Scots, is due to visit Edinburgh next month to give a fundraising speech for the Hunter Foundation, an education charity.

“Lots of people back in Scotland will be really looking forward to welcoming Barack Obama and hearing what he has to say,” Scottish First Minister Nicola Sturgeon said in a statement.

She recently confirmed that she would meet Mr. Trump if he visits, but her pro-Europe and pro-immigration stances are in stark contrast to the president’s views.

In Scotland, Mr. Trump could take the opportunity to visit his mother’s home village on Isle of Lewis. Mary Trump was born Mary MacLeod in a Scottish Gaelic-speaking family on the wind-swept island in the Atlantic and emigrated to the U.S. in the 1930s. She returned to Scotland many times before her death in 2000.

Isle of Lewis resident and lay preacher Derick Mackenzie, 53, has organized a campaign called Lewis for Trump. He and his five children will be out to greet the president if he comes, said Mr. Mackenzie.

“It is partially because of his family roots, being from Lewis, but also because I agree with his views on controlled immigration,” said Mr. Mackenzie. “I like certain aspects of his character and the way he does things. I like his approach — plain and simple common sense — which is very straight-talking. I think you can credit his mother with that. It’s great to see someone with local Lewis blood in the White House.”

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