“I’ve always seen Russia as the guardian at the gate, as the easternmost outpost of our people,” said Sam Dickson, a white supremacist and former Ku Klux Klan lawyer who frequently speaks at gatherings of the so-called alt-right, a far-right fringe movement that embraces white nationalism and a range of racist and anti-immigrant positions. “They are our barrier to the Oriental invasion of our homeland and the great protector of Christendom. I admire the Russian people. They are the strongest white people on earth.”

Fascination with and, in many cases, adoration of Mr. Putin — or at least a distorted image of him — first took hold among far-right politicians in Europe, many of whom have since developed close relations with their brethren in the United States. Such ties across the Atlantic have helped spread the view of Mr. Putin’s Russia as an ideal model.

“We need a chancellor like Putin, someone who is working for Germany and Europe like Putin works for Russia,” said Udo Voigt, leader of Germany’s National Democratic Party. That far-right group views Chancellor Angela Merkel as a traitor because she opened the door to nearly a million migrants from Syria and elsewhere last year.

“Putin is a symbol for us of what is possible,” Mr. Voigt said.

The Obama administration has accused Russian interests of meddling in the presidential campaign by spreading fake news and hacking into the computers of the Democratic National Committee and the emails of John Podesta, a leading figure in Hillary Clinton’s presidential campaign. But efforts by Russia, which has jailed some of its own white supremacist agitators, to organize and inspire extreme right-wing groups in the United States and Europe may ultimately prove more influential.

His voice amplified by Russian-funded think tanks, the Orthodox Church and state-controlled news media, like RT and Sputnik, that are aimed at foreign audiences, Mr. Putin has in recent years reached out to conservative and nationalist groups abroad with the message that he stands with them against gay rights activists and other forces of moral decay.