Despite the speculation, Principal Deputy White House Press Secretary Sarah Huckabee Sanders told reporters Tuesday that "while the president has the right to [fire Mueller], he has no intention to do so."
However, a close confidante of Trump's said Mueller's ousting is under consideration.
Chris Ruddy, an ABC News contributor and a longtime friend of Trump and the CEO of the conservative media company Newsmax, first told PBS in an interview Monday that the president is “considering, perhaps, terminating the special counsel, he’s weighing that option."
Ruddy told ABC News that he stands by his comments that the president is considering asking Rosenstein to terminate Mueller. “Trump is definitely considering it ... it’s not something that’s being dismissed,” Ruddy said.
According to a senior White House aide, there are no private conversations taking place about firing Mueller, and a White House spokesperson said Ruddy "speaks for himself."
Federal regulation states that the termination of a special counsel would have to come from the attorney general's office.
"The Special Counsel may be disciplined or removed from office only by the personal action of the Attorney General," the Code of Federal Regulation states.
"The Attorney General may remove a Special Counsel for misconduct, dereliction of duty, incapacity, conflict of interest, or for other good cause, including violation of Departmental policies. The Attorney General shall inform the Special Counsel in writing of the specific reason for his or her removal," it says.
In 1973 President Richard Nixon ordered his attorney general to fire Archibald Cox, the special prosecutor who was leading the Watergate investigation. The attorney general, Elliot Richardson, resigned rather than doing so, putting the onus on deputy attorney general William Ruckelshaus, who also chose to resign.
Solicitor general Robert Bork ultimately fired Cox.
In Mueller's case, Trump would have to ask Rosenstein to do the firing. If Rosenstein declined, the task would then fall to acting solicitor general Jeffrey Wall.
A number of key Republicans have publicly advised Trump to keep Mueller.
"I think we should let Bob Mueller do his work and get to the bottom of it, and get to the bottom of it quickly so that he can be vindicated, get to these things. Let’s not forget what this is originally all about. Russia is up to no good. Russia is trying to meddle into our elections," House Speaker Paul Ryan said today in an appearance on "The Hugh Hewitt Show."
"I don't think Trump should do anything but the congressional Republicans ought to look into it," Gingrich said.
Starr on Monday called Mueller "a great man" who is "honest as the day is long" and has a "great, great team of complete professionals."
What the Trump administration has said
During a press conference on May 18, the day after Mueller was named special counsel, ABC News' chief White House correspondent Jonathan Karl asked Trump if his appointment was the right move.
"Well, I respect the move, but the entire thing has been a witch hunt. And there is no collusion between certainly myself and my campaign, but I can always speak for myself -- and the Russians, zero," Trump responded.
ABC News' Adam Kelsey, Katherine Faulders, and Ben Siegel contributed to this report.