A secretive summit, an open invitation to visit the United States and now a letter hand-delivered by a sympathetic senator — as President Trump makes increasingly un­or­tho­dox overtures to Russian President Vladi­mir Putin, the GOP is facing a politically existential choice: cross the president and his allies, or accept that the party is warming to the Kremlin.

Trump has challenged traditional Republican antipathy toward Russia since taking office, even as his administration steps up sanctions against Moscow for various aggressive activities — most recently Wednesday’s announcement that the administration would increase punitive measures over the recent Russian nerve agent attack on a former Russian agent now living in Britain.

But while the president’s outreach to Putin has elicited a rebuke from many of the GOP’s fiercest national security hawks, it has also earned the tacit support of some Republicans, who in recent weeks have sidestepped party leaders to make their own diplomatic overtures to Moscow, as well as wide swaths of the Republican electorate, which polling has shown is starting to embrace Trump’s line on Russia.

The question is which faction of the GOP will prevail: those like Sen. John McCain (R-Ariz.), who advocate a clenched-fist approach toward the Kremlin, especially as the dispute over Russia’s interference in the 2016 election intensifies, or those endorsing Trump’s efforts to establish more friendly relations, despite the warnings of the national security community.

A visit to Moscow this week by Sen. Rand Paul (R-Ky.) marked the sharpest endorsement yet of Trump’s polarizing policy toward Russia. Paul punctuated his first day in the country by extending a surprise invitation to members of the Russian legislature to visit the United States in an attempt to establish a dialogue between the parliaments of both countries.

Paul is not the first lawmaker to go to Moscow in recent weeks: Last month, an eight-member Republican delegation visited. Members met with high-ranking lawmakers and Russian Foreign Minister Sergei Lavrov. But congressional leaders were quick to quash the idea that they are interested in hosting an exchange with Russian counterparts.

“Neither Congress nor the leader have invited any delegation from Russia to the Capitol,” said Don Stewart, a spokesman for Senate Majority Leader Mitch McConnell (R-Ky.), adding: “Senator Paul is the only one that I know who is discussing it.”

“That’s not something we’ve discussed,” said Brendan Buck, a spokesman for House Speaker Paul D. Ryan (R-Wis.).

Last month, Ryan and McConnell also said that Putin would not be welcome at the Capitol if he accepted Trump’s invitation to visit Washington.

Paul, a dogged noninterventionist, is known as something of a lone wolf on foreign policy. More recently, he has assumed the role of a one-man army fighting the many Republicans and Democrats who criticized Trump on Russia, particularly after the president’s summit with Putin in Helsinki, in which Trump appeared to take the Russian leader’s denials of election interference above the conclusions of the U.S. intelligence community.

On Wednesday, Paul said he delivered a letter to the Kremlin on behalf of Trump, calling it correspondence that “emphasized the importance of further engagement in various areas including countering terrorism, enhancing legislative dialogue and resuming cultural exchanges.”

Deputy White House press secretary Hogan Gidley argued that the White House had merely provided Paul with a letter of introduction at his request, including in it the “topics of interest that Senator Paul wanted to discuss with President Putin” — a meeting that did not occur.

But while Paul’s moves might lack a ready audience in Washington, some Republican voters in Kentucky applaud the effort.

“He wants to build off where Trump left off. I’m all for it. I think it’s long overdue,” said David Badgett, 44, a real estate broker. “What would be bad if we got along with Russians?”

“Any kind of diplomacy is good,” said Billy Williams, 34, a self-employed building contractor — who noted that he was dating a Russian woman. To him, Trump’s outreach to Putin “doesn’t seem weak; it seems diplomatic.”

A recent Gallup poll found that 40 percent of Republicans think Russia is either an ally or friendly toward the United States — nearly double the number of Republicans who thought so four years ago.

Several congressional experts pointed out that foreign policy only rarely ranks as a top issue for most U.S. voters, and few came into the present political moment with well-formed views on Russia — giving Trump plenty of latitude to sway public opinion.

“It’s not a topic that most Americans have been thinking a great deal about, and that’s what gives him such influence in this area,” Jack Pitney, a professor of political science at Claremont McKenna College, said of Trump. “For a certain number of Republican voters, Republicanism consists of whatever Trump stands for. So if Trump is more for Russia, they are too.”

That view was on striking display at Trump’s rally in Ohio this week, which some supporters attended wearing shirts that read: “I’d rather be a Russian than a Democrat.”

Even if GOP lawmakers don’t agree with such sentiments, they are not immune to that level of tribalism in a Republican base “that is in lockstep with Trump,” American Enterprise Institute congressional expert Norm Ornstein argued.

“To go against him, other than in a tweet here or there, risks being attacked by or shunned by them,” Ornstein continued. “When some of those who are clearly uneasy about [Trump’s Russia policies] do speak up, it’s in more muted tones. And often, they don’t.”

Only McCain — who has been in Arizona since December receiving treatment for a serious form of brain cancer — has refused to pull punches when tangling with Trump over Russia, Ornstein argued. He pointed out that even those who have fiercely advocated stepping up sanctions, such as Republican Sens. Marco Rubio (Fla.), Bob Corker (Tenn.) and Lindsey O. Graham (S.C.), have occasionally “soft-pedaled” their message so as not to anger Trump — whose condemnations of Russian aggression have been murky at best.

With little time on the calendar before the midterm elections, it is increasingly unlikely that lawmakers will do anything to push back against the president’s diplomatic efforts — or take any legislative steps to force a harder line against the Kremlin. GOP leaders have all but tabled efforts to swiftly pass new sanctions against Russia, opting first to hold hearings, and over the past few weeks, all Republican lawmakers but one — Corker — voted against devoting new funds to election security efforts.

“Even for the senators who object to what the president is doing on Russia and continue to see Russia as a threat . . . the degree to which they feel limited in what they can do to push back is a big part of the dynamic that’s at play,” said Molly Reynolds, a congressional expert at the Brookings Institution. And with elections looming, Trump’s sway over the GOP electorate could make the price of crossing the president even higher.

“It’s a classic case of opinion leadership. The president is out there making a case about Russia, and typical Republicans are following,” she said.

Jim Higdon in Louisville contributed to this report.