“We cannot allow foreign governments to interfere in our democracy,” Representative Michael McCaul, a Texas Republican who is the chairman of the Homeland Security Committee and was considered by Mr. Trump for secretary of Homeland Security, said at the conservative Heritage Foundation. “When they do, we must respond forcefully, publicly and decisively.”

He has promised hearings, saying the Russian activity was “a call to action,” as has Senator John McCain of Arizona, one of the few senators left from the Cold War era, when the Republican Party made opposition to the Soviet Union — and later deep suspicion of Russia — the centerpiece of its foreign policy.

Representative Peter T. King, Republican of New York and a member of the House Intelligence Committee, said there was little doubt that the Russian government was involved in hacking the Democratic National Committee. “All of the intelligence analysts who looked at it came to the conclusion that the tradecraft was very similar to the Russians,” he said.

Even one of Mr. Trump’s most enthusiastic supporters, Representative Devin Nunes, Republican of California, said on Friday that he had no doubt about Russia’s culpability. His complaint was with the intelligence agencies, which he said had “repeatedly” failed “to anticipate Putin’s hostile actions,” and with the Obama administration’s lack of a punitive response.

Mr. Nunes, the chairman of the House Intelligence Committee, said that the intelligence agencies had “ignored pleas by numerous Intelligence Committee members to take more forceful action against the Kremlin’s aggression.” He added that the Obama administration had “suddenly awoken to the threat.”

Like many Republicans, Mr. Nunes is threading a needle. His statement puts him in opposition to the position taken by Mr. Trump and his incoming national security adviser, Michael Flynn, who has traveled to Russia as a private citizen for RT, the state-controlled news operation, and attended a dinner with Mr. Putin.

Mr. Nunes’s contention that Mr. Obama was captivated by a desire to “reset” relations with Russia is also notable, because Mr. Trump has said he is trying to do the same — though he is avoiding that term, which was made popular by Mrs. Clinton in her failed effort as secretary of state in 2009.

There are splits both within the intelligence agencies and the congressional committees that oversee them. Officials say the C.I.A. and the N.S.A. have not always shared their findings with the F.B.I., which they often distrust. The question of how vigorously to investigate also has a political tinge: Democrats on the Senate Intelligence Committee, for example, are pushing hard for a broad investigation, while some Republicans are resisting.