"You have to hope that common sense would prevail", Schiff said. Jeanne Shaheen (D-NH), Rosenstein confirmed that he believes he has authority over Mueller's hiring and potential firing, and that Attorney General Jeff Sessions has forfeited that authority due to his recusal.
A spokesman for the office of the special counsel declined comment.
Trump's allies have begun raising questions about special counsel Robert Mueller's impartiality - he's a former FBI director who has worked with fired FBI Director James Comey - and floating the idea that Trump might replace him.
Newsmax CEO Chris Ruddy tells Judy Woodruff of "PBS NewsHour": "I think he's considering perhaps terminating the special counsel".
The White House did not immediately respond to questions about Ruddy's claims.
White House officials sought to downplay Ruddy's comments.
"I will not reveal who my sources are, but I felt confidence in saying it and it was consistent in what the president's own attorney said", Ruddy said in a Tuesday interview on MSNBC.
A person close to Trump's legal team said earlier Tuesday that there have been no discussions about firing Mueller with the legal team. The person demanded anonymity to discuss strategy on the sensitive matter.
Ruddy told CNN on Tuesday that Mueller and Trump "had a private conversation" and suggested it was "unethical" before the former Federal Bureau of Investigation director was named special counsel. That appeared to include Gingrich, who until recently expressed support for Mueller's appointment on the grounds that it would alleviate pressure in the Russian Federation affair.
Let's also recall that despite the media's tunnel vision on Trump, the wider Russian Federation investigation is examining a much wider array of issues and questions pertaining to our geopolitical adversary's malicious meddling efforts in the 2016 election.
Interior head suggests reducing Bears Ears National Monument
Trump's executive order authorized Zinke to review federal lands designated by Presidents Bill Clinton, George W. Utah rancher Zeb Dalton was among monument critics who wanted Zinke to recommend rescinding the entire monument.
But Stephen Gillers, a New York University professor who specializes in legal and judicial ethics, said the Mueller interview with Trump presented "no conflict whatsoever".
It's true that the ex-FBI director is amassing a team that not only includes top investigative talent but also has a partisan tilt: Jeannie Rhee, Michael Dreeben, Andrew Weissmann and James Quarles are renowned in their fields and have made repeated donations to Democrats.
Trump has repeatedly denied colluding with Russian Federation during the presidential campaign, and Comey and others have testified that there is no evidence to the contrary.
House Speaker Paul Ryan, however, did say he would be "surprised" if Trump fired Mueller.
But during an afternoon hearing, Sessions suggested to the Senate Intelligence Committee that Mueller could be trusted to carry out a fair investigation.
Under Justice Department regulations, firing Mueller would have to be done by the deputy attorney general rather than the president, although those regulations could theoretically be set aside. Sessions told senators on Tuesday that he'd recused himself because he was part of Trump's campaign.
"Time to rethink", he tweeted on Monday, citing Mr Mueller's hiring decisions and Mr Comey's admission that he instructed a friend to share with reporters notes he had taken of his private conversations with Mr Trump in order to force the appointment of special counsel.
Comey said he believed Mueller would look into Trump for obstruction of justice.
"I have talked to the president recently but we did not talk about the Mueller situation at all", Ruddy said.