But expressions of discontent with Mueller are bubbling up nonetheless.
Gingrich pointed out that some members of Mueller's team have in the past donated to Democratic politicians, which he said would make them biased against President Donald Trump.
Substitute anchor Kristen Welker, who led the interview about the alleged discussions about firing Mueller, replied, "Obviously, the legitimacy of the special counsel would be up for debate". Legal experts say that only a Justice official who has been confirmed by the Senate, as Bork had been, would have the authority to fire Mueller.
United States President Donald Trump is reportedly considering sacking Robert Mueller - the special independent counsel appointed to investigate allegations of Russian meddling in last year's U.S. election - according to a friend and close confidant to the President. "With respect to the subject, only the president or his attorneys are authorized to comment".
"I think the best advice is to let Robert Mueller do his job", Ryan told reporters Tuesday after a closed-door meeting with House Republicans.
That witness: James Comey, who replaced Mueller as Federal Bureau of Investigation director.
"I think he's weighing that option, " Ruddy said in an interview with the PBS NewsHour.
Peter Carr, a spokesman for Mueller, declined to comment on the issues Gingrich and others have raised.
Trump has boxed himself in a corner.
Firebrand pundit Ann Coulter urged the government to fire Mueller.
The Deputy Attorney General also said he would not follow any orders to fire special counsel Mueller unless there is a good reason and the orders are lawful.
After Trump fired James Comey in an attempt to quash the Russian Federation scandal investigation, he warned the former FBI director that he'd "better hope that there are no "tapes" of our conversations before he starts leaking to the press!"
He told Woodruff he thinks firing Mueller "would be a very significant mistake, even though I don't think there's a justification ... for a special counsel".
Still, his remarks prompted this reply on Twitter from Rep. Adam Schiff of California, the top Democrat on the Intelligence Committee, who said the committee would simply reappoint Mueller and then advised the president: "Don't waste our time".
Even so, Mr Gingrich said, Mr Trump will not seek to dismiss Mr Mueller because he is confident the probe will conclude he has done nothing wrong.
Barry Bennett, a GOP strategist who served as an adviser to Trump's campaign, said he believed it would be too damaging for Trump to try to remove Mueller, but that he had concerns about the appearance that the probe was being politicized.
New Hampshire Sen. Jeanne Shaheen, a Democrat, opened her line of questioning, in the hearing that was supposed to focus on the Justice Department budget, by asking if Rosenstein had seen any evidence of good cause to fire Mueller.
She said Mr Trump was not able to watch much of Mr Sessions' testimony, but he thought Mr Sessions "did a very good job" and was especially "strong" on denying any collusion between Mr Trump's campaign and Russian Federation.