Mr. Flake, of Arizona, announced his retirement on Tuesday. The senator, who has been outspoken against Mr. Trump for months, is the fourth prominent Republican to issue a stinging and recent rebuke against Mr. Trump’s presidency and style of governing.

Senate Republicans cannot afford to lose any support from their own party if they want to pass significant tax legislation that the president has promised for months.

Mr. Flake told CNN on Wednesday that he had decided not to run because he did not think he could be re-elected in a Republican primary race in Arizona. Mr. Flake would have faced Kelli Ward, a far-right former Arizona state senator.

“I couldn’t run the kind of race that I would be proud of and win in a Republican primary at this time,” Mr. Flake told CNN’s “New Day” on Wednesday. “The politics in that way has changed. You can be conservative on policy and it doesn’t matter, it seems, as much as being with the president or not criticizing him, even if you think he’s wrong.”

Mr. Corker, of Tennessee, would have been a heavy favorite for re-election in the Republican-tilted state. He told CNN on Wednesday that he was in a “commanding” position when he decided not to run in the 2018 race and that Mr. Trump’s assertions on Twitter about his poor chances were “just not true.” In September, Mr. Corker said his decision not to run again was difficult but that he was always upfront with his constituents that he saw himself serving only two terms.