Democratic Sen. Chris Van Hollen of Maryland, meanwhile, asked Rosenstein whether a direct order from President Trump would constitute "good cause" to fire Mueller. He described Mueller as operating independently from the Justice Department in his investigation.
Trump has made it clear that he is frustrated by the U.S. probe of Russian interference in the presidential election, which U.S. intelligence officials said was done to tilt the outcome in Trump's favor.
Ruddy also said Monday evening that Trump interviewed Mueller for the Federal Bureau of Investigation director position the day before he was named as special counsel. Rosenstein would be the one to fire Mueller, which he repeatedly assured senators Tuesday he would not consider without "good cause".
His testimony comes less than a week after Comey said during his own appearance before the Senate Intelligence Committee that he felt Trump had violated the normal separation of criminal investigations from White House oversight by urging him to end the investigation into Russia's activities.
Gingrich tweeted, "Republicans are delusional if they think the special counsel is going to be fair". Gen. Jeff Sessions rightly recused himself from any investigation connected with the election. It's unclear whether the president could override those rules and order Rosenstein to fire Mueller.
"I think he is considering perhaps terminating the special counsel", Ruddy said.
The revised ban, which Trump has said repeatedly now is a "travel ban", has been stayed by two appellate courts - the Fourth Circuit was the first, last month - and the administration has sought a review by the Supreme Court, and this injunction is headed for it as well. Rosenstein appointed Mueller as special counsel about a week later.
Lawmakers wanted Rosenstein to spell out the scope of Sessions' recusal, specifically with regard to the Russia investigation, which Sessions announced after reports emerged that he had not fully disclosed meetings with the Russian ambassador to the United States.
Rosenstein continued to defend the investigation and the Department of Justice's oversight, saying "I can assure you we're going to do the right thing and defend the integrity of the investigation". Brian Schatz, D-Hawaii.
Mr Sessions was revealed to have met the Russian ambassador twice previous year, meetings that he did not indicate during the Senate confirmation hearing approving his appointment. He is at the Senate committee and goes to the House one at 2 p.m. ET.
"Does Trump think Mueller needs this paycheck to meet a vehicle payment or something?" Deace explained that the best way for President Trump to move the country forward is to govern well.
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.
Robert Mueller was named special prosecutor for Russian Federation probe.
Sessions was supposed to testify before the appropriations committee, as past attorneys general have, but on Saturday, Sessions wrote the chairmen of both committees and said he was sending his deputy attorney general to testify in his place.