The White House and Trump's lawyers have in the past repeatedly said Trump was not planning to fire Mueller. Legislative leaders including House Speaker Paul Ryan, R-Wis., have said they've "received assurances" that Mueller's job is secure.
Ty Cobb, the White House lawyer dealing with the special counsel investigation, has not responded to NPR requests for comment on Mueller's future since the Cohen raid.
Under Justice Department rules, a special counsel can only be fired for "good cause," and some legal experts say that to remove Mueller, Trump would have to go through Rosenstein, who appointed Mueller after Attorney General Jeff Sessions recused himself from investigations of the 2016 election.
Trump had harsh words for Sessions and Rosenstein on Monday, but Sanders would not say whether Trump would take any action to move them from their posts.
Trump reportedly sought to fire Mueller last summer, but White House counsel Don McGahn refused to go along. Trump dismissed that report as "fake news."
Democrats Call for Legislation
Trump's remarks on Monday prompted top Senate Democrats to renew their calls to pass bipartisan legislation to protect Mueller's job.
Senate Minority Leader Chuck Schumer, D-N.Y., said it "looks more and more like a possibility" that the president will move to fire Mueller and shut down the investigation.
"We Democrats are urging our Republican colleagues to work with us now to pass legislation to protect his investigation and head off a constitutional crisis that would be devastating to this country. Let's not wait until it's too late," Schumer said.
There are several bipartisan Senate bills intended to protect Mueller. Sens. Chris Coons, D-Del., and Thom Tillis, R-N.C., introduced legislation last August that would create a process to let a fired special counsel appeal the removal before a panel of three federal judges within 14 days of the firing.
If the judges ruled the firing was without cause, the special counsel would be reinstated immediately.
"It is long past time for us to move forward with this legislation," Coons said on Tuesday.
Republicans Sanguine
Republicans' posture has not changed since the last flare-up about Mueller's job security.
GOP leaders have maintained that Mueller's investigation should run its course without interference and that there is no need to move forward with legislation to protect Mueller because they do not believe the president will make any moves to fire him.
"It's still my view that Mueller should be able to finish his job, I think that's the view of most people in Congress," said Senate Majority Leader Mitch McConnell, R-Ky.
The political backlash that might follow a move by Trump to get rid of Mueller would be so intense that the White House has a powerful incentive not to attempt it, the thinking goes. Senate Judiciary Committee Chairman Chuck Grassley, R-Iowa, said Tuesday that Trump would be committing "suicide."
That's why, for McConnell, there is no need for Senate Republicans to pass a bill.
"I haven't seen clear indication yet that we needed to pass something to keep him from being removed because I don't think that's going to happen, and that remains my view," McConnell said Tuesday. "I don't think he's going to be removed from this office, he shouldn't be removed from the office, he should be allowed to finish the job."
Tamara Keith contributed to this story.
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