The latest Democratic efforts would require cooperation from Republicans, who control the Senate and the House, as both parties negotiate over must-pass legislation to fund the government. The deadline for passing the spending plan is Feb. 8.
Senator Mark Warner, Democrat of Virginia, said on Friday that Congress should now pass the legislation to protect the special counsel, urging his Republican and Democratic colleagues to work out any differences in approach to make it happen.
Mr. Warner, speaking on CNN, said that he could not think of any reason “why you wouldn’t want to pass that legislation if you respect the rule of law.”
“Our oath of office was to the Constitution,” he said. “Quite honestly, history is going to judge how we act here.”
In August, Senator Thom Tillis, Republican of North Carolina, proposed legislation that would, among other things, require a court hearing if Mr. Mueller were fired. If the firing were found to be improper, the judges could reinstate him.
“Special Counsel Mueller is a career professional respected by both sides of the aisle for good reason,” Mr. Tillis’s office said in a statement on Friday. “He should be able to do his job without elected officials trying to score cheap political points for their own partisan gain.”
Democratic aides said they hoped Republicans would join their call to include such protections for the special counsel in budget negotiations. They said lawmakers would be working during the next several days to merge the best parts of previous legislation that sought to do so.
One measure, sponsored in part by Senator Lindsey Graham, Republican of South Carolina, would require a judge’s review to ensure a special counsel is fired for cause and not for political reasons.
The proposal introduced by Mr. Tillis would also require a Senate-confirmed official at the Justice Department to discipline or fire a special counsel. That measure was introduced in the summer, when Mr. Trump had privately and publicly disparaged his attorney general, Jeff Sessions. Some feared Mr. Trump would fire top officials at the Justice Department until someone carried out his orders to fire the special counsel.
“This remarkable report makes scarily clear that we need this protection right away for the special counsel,” Senator Richard Blumenthal, Democrat of Connecticut, said on Friday. “It’s necessary now to send a signal to the president that political interference and firing the special counsel simply is totally unacceptable, and there’s bipartisan unanimity that it would be unconscionable and unacceptable.”
Mr. Blumenthal said that some Republican senators have told him that they support such protections. He did not name them.
Even as some Republicans have tried to discredit the Russia inquiry, some senior members of the party have said that they would not support the firing of Mr. Mueller.
A spokesman for Speaker Paul D. Ryan, of Wisconsin, said on Friday that Mr. Ryan’s position had not changed since he said in June that Mr. Mueller should be left alone to do his job.
And a spokesman for Senator John Cornyn of Texas, the No. 2 Republican in the Senate, said that Mr. Cornyn still believed that it would be a “mistake” to fire the special counsel.
Representative Tom Cole, Republican of Oklahoma, reiterated Friday that Republicans in the House would oppose any attempt by Mr. Trump to fire Mr. Mueller.
“It would just be a political firestorm, and you would just be presumed guilty whether you are or not,” said Mr. Cole, who had hired Mr. McGahn to be his counsel when he was the chairman of the National Republican Campaign Committee. “It’s easier to rebut the outcome than deal with suspicions about why you fired somebody. It would be taken as a confession of guilt. And every analogy would be Nixon and the Saturday Night Massacre.”
Mr. Trump denied on Friday that he had ordered Mr. Mueller’s firing and called reports of the episode “fake news.” His comments came during a trip to Davos, Switzerland, where he was attending the World Economic Forum, a gathering of world leaders and global business executives.
The Times report was based on four people who were told of the matter. On Thursday, Ty Cobb, who manages the White House’s relationship with Mr. Mueller’s office, declined to comment.
The June episode could emerge as an important part of Mr. Mueller’s inquiry, part of which is looking into whether Mr. Trump or anyone in the White House or associated with his campaign obstructed justice by trying to impede investigators examining the possibility of campaign-related collusion with Russia.
Mr. Trump’s order to fire Mr. Mueller came a month after the president dismissed the F.B.I. director James B. Comey, later citing the Russia inquiry as the reason for the decision. At the time, Mr. Comey oversaw the F.B.I.’s investigation into collusion with Russia during the election. The firing of Mr. Comey in May directly led to Mr. Mueller’s appointment.
Mr. Trump’s denial of the June episode echoes repeated statements by the president and other White House officials that he had never considered firing the special prosecutor.
“I haven’t given it any thought,” Mr. Trump told reporters in August. “Well, I’ve been reading about it from you people. You say, oh, I’m going to dismiss him. No, I’m not dismissing anybody.”
John Dowd, the president’s personal lawyer, said that same month that firing Mr. Mueller had “never been on the table, never.”
But four people, who spoke on the condition of anonymity because they did not want to be identified discussing a continuing investigation, said Mr. Trump ordered the firing, citing what he believed were three reasons that Mr. Mueller has a conflict of interest that should prevent him from leading the Russia investigation.
Those included claims about a disputed payment of fees by Mr. Mueller at Trump National Golf Club in Sterling, Va.; the fact that Mr. Mueller worked at the same law firm that represented the president’s son-in-law; and Mr. Mueller’s interview with the president to be F.B.I. director before he was appointed to be the special counsel.