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WASHINGTON — Congressional Republicans face a vexing dilemma with the impending presidency of Donald J. Trump: Will they maintain the tough line on Russia that has been central to their foreign policy for decades, or cede that ground to Democrats?

For decades during the Cold War, Republicans tried to claim the hawkish mantle when it came to confronting the Soviet Union. Vice President Richard M. Nixon famously squared off against Premier Nikita Khrushchev in 1959, and years later President Ronald Reagan cast the Soviets as an “evil empire.”

Reagan made that kind assertiveness central to his foreign policy, and he is credited by many with hastening the downfall of the Soviet Union, the most persistent and formidable adversary of the United States in the last 60 years. And Reagan disciples today in the Republican Party, including Vice President-elect Mike Pence, are many.

Reagan helped to frame the template for an American foreign policy that promulgated democracy around the world and curbed what has often been called Russian adventurism.

Now Republicans will have to reconcile that party catechism with their vote on Mr. Trump’s selection as secretary of state, Rex W. Tillerson, who is the chief executive of Exxon Mobil and a longtime friend of President Vladimir V. Putin of Russia.

Mr. Tillerson, who has described his relationship with Mr. Putin as close, was once presented with the Russian Order of Friendship, one of the highest honors a foreigner can receive. Mr. Trump’s selection of him drew strong condemnation from Democrats and a cool reception from a handful of Republicans like Senator John McCain, Republican of Arizona, a longtime leader against Russian aggression.

“Russia is going to be the central litmus test for United States policy,” said Heather A. Conley, a Russia expert at the Center for Strategic and International Studies. “Our our allies and our adversaries are watching this very closely, and obviously the names of the cabinet positions are being scrutinized that much more closely.”

Senate Republicans — including Senator Bob Corker of Tennessee, the chairman of the Foreign Relations Committee — who have long been critical of Mr. Putin and of President Obama’s attempt to “reset” relations with Moscow, have praised Mr. Tillerson.

“Mr. Tillerson is a very impressive individual and has an extraordinary working knowledge of the world,” Mr. Corker said. That view was echoed by Senator Mitch McConnell of Kentucky, the majority leader.

The laudatory response from many Republicans over a choice that a year ago, on paper at least, might have appalled them demonstrates a strong desire to begin a new administration aligned with Mr. Trump.

It is the same dynamic that has prevented a larger outcry from congressional Republicans overs revelations that Russia interfered with the presidential election. They fear they could appear aligned with Democrats in raising questions about the election’s legitimacy. While congressional leaders called for investigations into possible tampering, they stopped short of ordering expansive efforts like a select committee.

At the same time, a majority of Republicans are overjoyed with Mr. Trump’s other cabinet picks — staunch conservatives in the world of education, health care and law enforcement — and are likely to accede to the president-elect’s choice for the nation’s chief diplomat.

The other selections are “draft picks for conservatives who have been looking to reform those departments for years,” said Kevin Madden, a former adviser to Mitt Romney, who was passed over for secretary of state. “Those cabinet picks have certainly helped build up some of that political capital.”

Still, the contrast from recent years is striking.

In 2012, when Mr. Romney was running for president, he called Russia the “No. 1 geopolitical foe,” a position echoing decades of Republican thinking. He was derided by President Obama, his opponent.

Mr. Corker and others have joined the most robust voices on Capitol Hill in calling for sanctions on Russia, a position that would seem to put them at odds with Mr. Trump and Mr. Tillerson. He and 20 Senate Republicans tried in 2014 to push through severe new sanction triggers against the nation, and praised Mr. Obama when he imposed them on Russia for destabilizing Ukraine.

Both of the last two major defense bills authorized funding for security assistance to Ukraine, including lethal assistance the Obama administration has refused to provide.

There have been notable exceptions to the Republicans praising Mr. Tillerson. Senator Marco Rubio, Republican of Florida, said this week: “I have serious concerns about his nomination. The next secretary of state must be someone who views the world with moral clarity.”

Senator Lindsey Graham, who often aligns with Mr. McCain, said in a statement that he expected “the U.S.-Russian relationship to be front and center in his confirmation process.” Two other Republicans, Senators James Lankford of Oklahoma and Ben Sasse of Nebraska, reacted to the news as if they had been presented with their sixth choice on a lunch menu of 10 items.

“Senator Sasse has been outspoken against Russia’s recent aggressions,” said his spokesman, James Wegmann. “He also looks forward to diving into every nominee’s record.”

The burden will fall to Mr. Tillerson, and perhaps Mr. Trump, to persuade Mr. Rubio and enough other Republicans that he shares their views on Russia, his friendship with Mr. Putin notwithstanding.

The process, Ms. Conley said, may well provide clues to allies and adversaries about where the United States is headed under Mr. Trump and a Republican-controlled Congress in what has been a fraught relationship with Russia.

“The question is: Is the United States willing to accommodate the Russian annexation of territory, the invasion of its neighbors and its indiscriminate bombing of civilians in Syria, or are we willing to defend principles and rules that go back to the end of the Second World War?” Ms. Conley said. “If the U.S. walks away from these principles, other countries such as Russia will step into the breach and trample on the very rules that keep Americans safe.”