If Mr Trump instructs Mr Rosenstein to dump Mr Mueller, it would evoke memories of 1973, when two top Justice Department officials, Mr Elliott Richardson and Mr Bill Ruckelshaus, resigned rather than obey then President Richard Nixon's order to fire Mr Archibald Cox, the special prosecutor conducting the Watergate investigation.
High-profile supporters of President Donald Trump have started to turn on Mueller as he builds his legal team, raising questions about the former Federal Bureau of Investigation director's impartiality and suggesting he can not be trusted to lead the probe.
Rosenstein's letter critical of Comey's handling of the Hillary Clinton email server investigation served as the Trump White House's first reasoning for firing Comey - though Trump later undercut that explanation by saying he fired Comey in hopes of ending the Russian Federation probe. South Carolina Republican Lindsey Graham, who is unafraid to tangle with Trump, said firing Mueller would be "a disaster, " since the special counsel had done nothing to be fired.
Mr Sessions was revealed to have met the Russian ambassador twice past year, meetings that he did not indicate during the Senate confirmation hearing approving his appointment.
Ruddy, an early donor to Trump's presidential campaign, was reportedly at the White House on Monday.
President Trump's supporters have a history of manufacturing story lines to undermine opponents.
Mueller has wide authority to look into matters related to Russian election meddling, including collusion with Trump operatives, financial links between Trump and Russia and whether the president tried to obstruct the inquiry.
U.S. attorney general Jeff Sessions has said that any suggestion of collusion between him and Russian Federation is "an appalling and detestable lie", as he appeared before the Senate Intelligence Committee in Washington. If there were good cause, I would consider it.
He says he is confident that Mueller will have "the full independence he needs" to investigate thoroughly.
Neither Trump nor Attorney General Jeff Sessions has offered public support for Mueller.
Ruddy emphasized that he thinks firing Mueller would be "a significant mistake" but also that he thinks there is no justification for a special counsel in the first place.
Democrats, reacting to the chatter, said that if Trump fired Mueller they'd try to enact an independent counsel statute so they could appoint him.
Still, Chris Ruddy, the CEO of Newsmax and a Trump friend, suggested the president was already thinking about "terminating" Mueller from his position as special counsel. "If there were not good cause, it wouldn't matter to me what anybody says", Rosenstein added.
Democrat Adam Schiff, a high-ranking member of the House intelligence committee, said if Trump were fired, Congress would bring him right back as independent counsel. "If there are other witnesses or subjects, we don't want them hearing what other people may have said about it".
As attorney general, Sessions is unlikely to answer in detail questions about conversations he's had with Trump.
A source close to Trump said the president is being counseled to avoid firing the special counsel.
Meanwhile, House Speaker Paul Ryan urged the White House to let Mueller's investigation play out.
According to the report, Trump has told staff, visitors and advisers over the last week that he thought Mueller was part of a "witch hunt" against him.