The White House, however, said Trump and Ruddy have never discussed the issue.
Ruddy said the ethics behind Mueller's presence are concerning considering how he was appointed to be special counsel.
"It's a mistake to pretend that this is going to be some neutral investigation", Gingrich said.
If he pulls the trigger on Mueller, Frum said, Trump should expect the same response Congress offered when Nixon ordered first Attorney General Elliot Richardson, then Deputy Attorney General William Ruckelshaus and finally Solicitor General Robert Bork to fire special prosecutor Archibald Cox.
"In this kind of environment, I don't give the benefit of the doubt to somebody who can only hire Democrats, but claims that we ought to trust him", Gingrich said.
Rosenstein may be asked to address the issue when he speaks at a Senate subcommittee hearing Tuesday morning.
"I won't mince words, " Leahy told Rosenstein on Tuesday. He added that "if there were good cause, I would consider it".
Multiple House Republicans said that there was no discussion about the special counsel at the weekly GOP meeting. Richardson refused and resigned, as did Deputy Attorney General William Ruckelshaus.More news: Michael Jordan's Bulls? Showtime Lakers? The Golden State Warriors top them all
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Under questioning from Sen.
Rosenstein responded, "Senator, I'm not going to follow any orders unless I believe those orders are lawful and appropriate orders".
The responsibility for appointing (or theoretically firing) Mueller lies with Rosenstein because his boss, Attorney General Jeff Sessions, recused himself in March from all investigations involving Russian Federation and the 2016 elections.
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. Susan Collins, R-Maine, asked Rosenstein what he would do if the president asked him to fire Mueller. Gen. Jeff Sessions recused himself from any dealings with the Russia probe on March 2 after his own dealings with Russian officials was disclosed.
White House spokeswoman Sarah Huckabee Sanders said via email, "Chris speaks for himself". An order from the president would not necessarily qualify, he said. "I have confidence in Bob Mueller". I called for it some months ago.
In a Tuesday hearing on the Justice Department budget for the 2018 fiscal year, Sen. Should Rosenstein choose to resign, then the decision would be in an order of the sequence of command at the Justice Department.
The deputy attorney general's testimony comes after Trump friend and Newsmax CEO Chris Ruddy said Monday that Trump was considering "terminating" the special counsel who is overseeing the Justice Department's wide-ranging inquiry into possible collusion between the Trump campaign and Russian officials. The attorney general is expected to be closely questioned about his contacts with Kislyak and his role in last month's firing Federal Bureau of Investigation director James Comey.
Schiff says that if Trump fires Mueller, Congress might name its own independent counsel to investigate the case. They will also ask about his decision to recuse from an investigation into possible ties between Moscow and associates of President Donald Trump.
But both appearances, however, are being overshadowed by Sessions' public appearance at a Senate intelligence committee hearing.
Lawmakers for weeks have demanded answers from Sessions, particularly about meetings he had last summer and fall with the Russian ambassador to the United States.