The authority to appoint a special counsel fell to Rosenstein, after Attorney General Jeff Sessions recused himself from the Russia case because of his failure to disclose during his January confirmation hearing pre-election meetings he had Russian ambassador Sergey Kislyak.
Mr Trump is yet to comment publicly on the matter and White House spokesman Sean Spicer has told Washington media Mr Mueller's sacking is unlikely.
President Donald Trump has "no intention" of firing special counsel Robert Mueller who is investigating the alleged Russian meddling in last year's U.S. presidential polls, the White House has said.
Session will also testify before the House later Tuesday, where he is also scheduled to discuss the department budget.
Some of Mr Trump's allies have begun questioning Mr Mueller's credibility as the investigation intensifies. Ruddy later confirmed the comment to CNN. "I know Bob Mueller, and I have confidence in Bob Mueller", Ryan said.
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The leaders of the Senate intelligence committee say In a statement issued Wednesday that they "look forward to future engagements" with Robert Mueller.
According to Ruddy, Mueller was a candidate to replace James Comey as Federal Bureau of Investigation director and had met privately with Trump on that subject. On Monday, speculation was stoked that Trump was considering terminating Mueller, after a confidante of the president suggested it was under consideration. "I don't think the Chairman will invite the President, a man he campaigned for, because the role of the House Judiciary Committee right now is to protect the President at all costs; shielding the President from tough questions", he said.
"I think that he should be more cognizant of making sure that the people who come to work with him are non-partisan, for his own credibility", Bennett said of Mueller.
"Time to rethink", he tweeted Monday, citing Mueller's hiring decisions and Comey's admission that he'd instructed a friend to share with reporters notes he'd taken of his private conversations with Trump in order to force the appointment of a special counsel.
Speaker of the House Paul Ryan, a Republican, said during a press conference on Tuesday, "I think the best vindication for the president is to let this investigation go on independently", meaning the president should not interfere with Mueller's work.
"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. He has always been widely respected by many in Washington from both sides of the aisle, with many lawmakers praising Deputy Attorney General Rob Rosenstein's pick.
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