“Let me say that I have the highest confidence in the intelligence community,” Mr. McConnell said, “and especially the Central Intelligence Agency. The C.I.A. is filled with selfless patriots, many of whom anonymously risk their lives for the American people.”

The top Republican in the House, Speaker Paul D. Ryan of Wisconsin, said he supported a continuing investigation by Representative Devin Nunes of California, the chairman of the House Intelligence Committee. In a statement, Mr. Ryan said: “As I’ve said before, any foreign intervention in our elections is entirely unacceptable. And any intervention by Russia is especially problematic because, under President Putin, Russia has been an aggressor that consistently undermines American interests.”

Congressional Republicans announced their support for inquiries after Mr. Trump railed for much of the weekend against the intelligence findings. But their remarks, especially Mr. Ryan’s, were far from fiery, reflecting both a fear of offending Mr. Trump, who has taken many positions against traditional Republican orthodoxy, and the Republicans’ belief that Democrats have selectively leaked intelligence information for political gain.

Critics from both parties are questioning Mr. Trump’s apparent choice of Rex W. Tillerson, the chief executive of Exxon Mobil, as secretary of state, particularly because of his longstanding business connections with Russia and his close relationship with President Vladimir V. Putin, whom he has known for two decades. Mr. Trump said in a Twitter post on Monday night that he would make a formal announcement on the job on Tuesday morning.

Senators Lindsey Graham of South Carolina and Marco Rubio of Florida, both Republicans, have expressed concern about the reports of cyberattacks, as have numerous Democrats. But Mr. Rubio, in an apparent reference to Mr. Tillerson, went a step further on Monday, writing on Twitter, “Being a ‘friend of Vladimir’ is not an attribute I am hoping for from a #SecretaryOfState.”