Rosenstein, who appointed Mueller in mid-May as special counsel overseeing the the investigation into whether the President Donald Trump's campaign colluded with Russian officials during the 2016 election, noted that under the special counsel regulations, only the attorney general has the authority to fire such an appointee.
The White House was quick to dismiss Ruddy's comments.
Schiff says that if Trump fires Mueller, Congress might name its own independent counsel to investigate the case.
Mueller's investigators have asked for information and will talk to Director of National Intelligence Dan Coats and National Security Agency Director Adm. Mike Rogers, a source familiar with the matter told CNN.
On Monday, a friend of Trump's, Newsmax CEO Chris Ruddy, claimed that Trump was "considering, perhaps, terminating the special counsel" - an action that would ultimately have to be made by Rosenstein.
"I appointed him; I stand by that decision, " he said.More news: Investigation Continues Into Congressional Baseball Practice Gunman
More news: On Twitter, Trump decries obstruction probe related to 'phony' Russian Federation story
More news: Trump Says Obstruction Investigation Based On 'Phony' News
"You're creating a debate that's not happening", said House Majority Leader Kevin McCarthy Tuesday morning, responding to a reporter's question about the rumors.
At a June 13 hearing at a House of Representatives panel, Deputy Attorney General Rod Rosenstein refused to say who asked him to write a memo justifying Mr. Comey's firing.
It is today been described as a "crime scene" following revelations the probe into Russia's interference in the U.S. election has been widened to include allegations President Donald Trump attempted to obstruct justice. Comey testified in a Senate hearing last week that he believed he was sacked "because of the Russian Federation investigation".
For many, the prospect of Mueller's firing immediately brought to mind the 1973 dismissal of special prosecutor Archibald Cox during the Watergate scandal.
Justice Department spokesman Ian Prior declined to comment on the possibility of Trump seeking to remove Mueller. Until now, it was examining Russia's alleged efforts to tilt last November's presidential election Mr Trump's way and whether the winner's campaign was involved in any way. Read Also: Trump faces new lawsuit over business empireSeveral lawmakers came out in support of Mueller.
Under Department of Justice rules, it is actually up to the deputy attorney general to make that call, though the president could bypass that protocol.