But House Majority Leader Kevin McCarthy (R-Calif.) said there was no need to pass such a measure, as he defended how the president and his team have navigated Mueller's probe into Russian meddling in the 2016 presidential election.
The GOP discord came just days after the revelation that Trump sought Mueller's ouster last June, prompting Democrats to make a renewed pitch for Congress to shore up the special counsel's standing. It underscored the growing split in the Republican Party between Trump loyalists and others who are becoming increasingly concerned with the president's actions.
That rift presents a challenge for lawmakers hoping to place new limits on Trump's authority. Republicans control both chambers of Congress, and many in the party have been reluctant to take a hostile posture toward the president, who holds considerable influence over the conservative base despite his low approval ratings nationally.
"I have got legislation protecting Mr. Mueller. And I'll be glad to pass it tomorrow," Graham said on ABC's "This Week with George Stephanopoulos." He was referring to a proposal he unveiled last August with Sen. Cory Booker (D-N.J.) to require a panel of judges to review any decision to fire a special counsel before it is final.
"Everybody in the White House knows it would be the end of President Trump's presidency if he fired Mr. Mueller," Graham said.
Sens. Thom Tillis (R-N.C.) and Christopher A. Coons (D-Del.) have offered a similar plan. Collins said on CNN's "State of the Union" that adopting some version of their collective ideas could be helpful.
"It would certainly not hurt to put that extra safeguard in place, given the latest stories," she said. Late last year, Collins was cooler toward the idea of moving to shield Mueller.
In the House, where GOP lawmakers have tended to align themselves more closely with Trump, McCarthy showed no appetite for moving ahead with those kinds of bills.
"I don't think there's a need for legislation right now to protect Mueller," McCarthy said on NBC's "Meet the Press." He said Trump and his team "have fully cooperated" with the investigation.
McCarthy expressed confidence in Mueller but questioned the motivations of some others in the FBI and Justice Department who have been involved in the probe, citing the revelations of politically charged texts disparaging Trump. Graham voiced a similar sentiment.
Trump sought the firing of Mueller last June and backed off only after White House Counsel Donald F. McGahn threatened to resign, two people familiar with the episode confirmed on Thursday.
White House legislative affairs director Marc Short was asked directly on "Fox News Sunday" whether Trump wanted to fire Muller last summer. He responded carefully.
"I'm not aware the president ever intimated that he wanted to fire Robert Mueller," Short said. He declined to say what Trump would do if Congress acted to make it more difficult for him to get rid of the special counsel.
"I don't know, hypothetically," he said.
Democrats have advocated proceeding with the proposals to reinforce Mueller's standing, even as many Republican lawmakers and aides have shown little urgency about acting. Senate Minority Leader Charles E. Schumer (D-N.Y.) has said Democrats will try to add the protections during the government spending negotiations.
Not all Democrats are embracing that tactic. Asked Sunday on CNN whether it would be a good idea, Sen. Joe Manchin III (D-W.Va.), a centrist facing reelection this year, said that it would be "premature for us to go down that road."