WASHINGTON — The White House is trying to soothe Republicans who say the party might have fared better on Election Day if President Bush had not waited until after the vote to oust Defense Secretary Rumsfeld.

“You could argue that either way, of what political effect an earlier decision on Secretary Rumsfeld would have had. But it doesn’t matter,” White House chief of staff Josh Bolten said yesterday.

“The president correctly decided that this decision does not belong in the political realm. And a decision as important as your secretary of defense should not be made based on some partisan political advantage. It would send a terrible signal to our troops, to our allies, even to our enemies,” Mr. Bolten said.

A former House speaker, Newt Gingrich, has suggested that if Mr. Bush replaced Mr. Rumsfeld two weeks before the election, voters would not have been as angry about the unpopular Iraq war. Republicans would have gained the boost they needed, according to Mr. Gingrich, to retain their majority in the Senate and hold onto 10 to 15 more House seats.

The outgoing chairman of the Senate Judiciary Committee, Senator Specter, a Republican of Pennsylvania, had a similar assessment.

Mr. Bush should have removed Mr. Rumsfeld “as soon as he had made up his mind. And that’s a hard thing to calculate. But it’s highly doubtful that he made up his mind between the time the election returns came in on Tuesday and Wednesday when Rumsfeld was out.”

“And if Rumsfeld had been out, you bet it would have made a difference,” Mr. Specter said. “I’d still be chairman of the Judiciary Committee.”

The same thought occurred to veteran Rep. Clay Shaw, a Republican of Florida, who was on the verge of becoming chairman of the House Ways and Means Committee. After the election, Mr. Shaw said if Mr. Rumsfeld had stepped down before last Tuesday, Mr. Shaw and other Republicans might have won.

“It could have made a difference in who is running the Congress,” Mr. Shaw said.

Mr. Bush said in an Oval Office interview with the Associated Press and other reporters on November 1 that he expected Mr. Rumsfeld and Vice President Cheney to stay in his administration until the end. A day after Democrats triumphed, Mr. Bush acknowledged he misled the reporters because he “didn’t want to inject a major decision about this war in the final days of a campaign.”

Mr. Bolten explained that when Mr. Bush spoke to the reporters, the president had begun seeking a replacement for Mr. Rumsfeld — but had yet to decide on a successor.

“The president was not going to replace Secretary Rumsfeld unless he was confident that he had a very strong replacement available to him to put in place,” Mr. Bolten said.

More important, Mr. Bolten said, was that Mr. Bush’s misdirection to the press was justified by military need.

“If he had said something other than what he said, if he had been equivocal about his support for Secretary Rumsfeld, that would have started an outbreak of then warranted speculation about Secretary Rumsfeld’s tenure,” Mr. Bolten said. “It would have undermined Secretary Rumsfeld’s ability to lead the military in a time of war.”

White House counselor Dan Bartlett said he did not buy the idea that the timing of the resignation would have made a difference with voters — but allowed that the president was concerned with appearances.

“Think about the signal it would have sent two weeks before the election if President Bush, desperate to change political polls, would have jettisoned his secretary of defense. It would have looked desperate,” Mr. Bartlett said.

“It would have looked like it was made based on political motivations, not on the security interests of our country,” he said. “And I think that would have weakened the president and Republican support going down the stretch of this campaign.”

Mr. Rumsfeld’s departure was a turning point in the administration’s Iraq policy. Messrs. Bolten and Bartlett, making the rounds of yesterday’s talk shows, said Mr. Bush will now welcome new ideas about the war, even from Democrats he once called soft on terrorism.

Some Democrats raised concerns about Mr. Bush’s choice of former CIA Director Robert Gates to succeed Mr. Rumsfeld. The questions revolve around what Mr. Gates knew during the Iran-Contra affair, which involved two Reagan White House covert operations: selling arms to Iran to free American hostages in Lebanon, and supplying arms to the Contra guerrillas fighting the leftist government of Nicaragua.

“I haven’t decided what I’m going to do with Mr. Gates,” the incoming chairman of the Senate Armed Services Committee, Senator Levin, a Democrat of Michigan, said.

Messrs. Bolten and Levin spoke on ABC’s “This Week.” Mr. Specter appeared on CNN’s “Late Edition.” Messrs. Bartlett and Dean were on ” Fox News Sunday.”