Trump friend Chris Ruddy, the CEO of Newsmax, has gone so far as to suggest the president was already thinking about "terminating" Mueller. During the House hearing, New York Representative Nita Lowey asked Rosenstein if he had discussed former FBI Director James Comey's termination with the White House, and if anyone directed him to write his recommendation of Comey's dismissal.
"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.
As special counsel Robert Mueller builds his legal team, Mr Trump's allies have begun raising questions about the former Federal Bureau of Investigation director's impartiality, implying he can not be trusted to lead the probe.
Until now, Mueller had drawn widespread praise from Republicans and Democrats alike.
The White House appeared blindsided by it, but took hours to address it, triggering speculation it was, one, a plan indeed and Moody was testing the waters for Trump, and two, it was an attempt by a anxious administration to intimidate the counsel and the formidable team of lawyers he was putting together.
Sean Spicer, the White House press secretary, said Ruddy "never spoke to the president regarding this issue".
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.
Gashed Fitzgerald reaches port as search for sailors continues
That could account for the missing sailors, who may have been on watch on deck or even just relaxing in the night air, he said. The container ship's left bow was dented and scraped, but it did not appear to have sustained any major structural damage.
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". In a volley of hostility, last week media personality Rush Limbaugh took to the radio endorsing the suggestion for Trump to fire Mueller. It has been suggested that Ruddy based his judgement on comments made over the weekend by members of Trump's personal legal team.
The news comes on the eve of Attorney General Jeff Sessions testifying before the Senate Intelligence Committe on the Russian Federation investigation.
But it has today been described as a "crime scene" following revelations the probe into Russia's interference in the United States election has been widened to include allegations President Donald Trump attempted to obstruct justice.
Gingrich and Ruddy said it would be a "mistake" for Trump to remove Mueller, although both said they see the probe as unnecessary. "Can you just tell me, what is the allegation against the President or any of his officials?" The newspaper noted Mueller had requested interviews with CIA Director Dan Coats, National Security Agency chief Michael Rogers and Richard Ledgett, the former NSA deputy director. "If there were not good cause, it wouldn't matter to me what anybody says".
The former House speaker Newt Gingrich, a long-time supporter of Mr. Trump's, said in a tweet on Monday that it was time to "rethink" Mr. Mueller's ability to be fair.
Attorney General Jeff Sessions also is scheduled Tuesday to testify before the Senate Intelligence Committee, a venue where the possibility of firing Mueller could arise.