Susan Collins, R-Maine, asked Rosenstein during an open Senate Appropriations subcommittee hearing what he would do if Trump asked him to fire Mueller.
Trump also had other issues on his mind on Tuesday as reflected in his tweets, the Russian Federation probe among them, as he sought to deflect attention towards Hillary Clinton and the investigation into her use of a private email server that came up during fired FBI director James Comey's testimony last week.
Republicans in Congress are also shaking off the notion that Trump could fire Mueller.
Previously, a close friend of Trump and the CEO of the conservative media company Newsmax Chris Ruddy told "PBS NewsHour" in an interview Monday that the president is "considering, perhaps, terminating the special counsel, he's weighing that option." Mr.
"With respect to this subject, only the president or his attorneys are authorized to comment", he continued.
Trump can not directly dismiss Mueller.
On Monday, rumors emerged that President Donald Trump is allegedly considering firing Robert Mueller, the special prosecutor who was appointed to investigate the Trump campaign's alleged ties to Russian Federation.
He also declared it a "detestable and appalling lie" to suggest he was aware of or took part in any collusion between Russian Federation and the election campaign that sent Mr Trump to the White House. "I personally think it would be a very significant mistake". Rosenstein would be the one to fire Mueller, which he repeatedly assured senators Tuesday he would not consider without "good cause".
So as deputy attorney general, Rosenstein would have the power to remove Mueller from his position - as long as he had a good reason to do that and put it in writing.
Some have compared Trump's firing of then-FBI Director James Comey in May to the storied "Saturday Night massacre" that occurred during Richard Nixon's presidency. Trump himself started, declaring himself one of the most accomplished leaders the United States has ever seen. The attorney general, Elliot Richardson, resigned rather than doing so, putting the onus on deputy attorney general William Ruckelshaus, who also chose to resign. But Rosenstein, too, may ultimately have to hand off oversight given his role in Trump's decision to fire Comey.
A key takeaway: If the president wants to fire Mueller, he'll have to fire Rosenstein first. If Rosenstein declined, the task would then fall to acting solicitor general Jeffrey Wall. Let's not forget what this is originally all about. Ruddy later confirmed the comment to CNN.
Both Gingrich and Ken Starr were closely involved in the investigation and impeachment of former President Bill Clinton.
"I'd be insincere if I said it wasn't a concern that the president would try to do it anyway", the aide added.
Ruddy argued that Mueller should not have taken the job of special counsel because he was under consideration for the other position, and had had the private meeting with the president for the Federal Bureau of Investigation director appointment. "You are witnessing the single greatest WITCH HUNT in American political history - led by some very bad and conflicted people!"
However, White House press secretary Sean Spicer denied Ruddy has spoken to the President about the issue.