If Trump instructs Rosenstein to dump Mueller, it would evoke memories of 1973, when the two top Justice Department officials, Elliott Richardson and Bill Ruckelshaus, resigned rather than obey President Richard Nixon's order to fire Archibald Cox, the special prosecutor conducting the Watergate investigation.
Mr Trump's friend, Mr Christopher Ruddy, chief executive of Newsmax Media, said on the Public Broadcasting Service's NewsHour programme that the President was "considering, perhaps, terminating the special counsel".
Sessions lent his support to the special counsel, Robert Mueller, who is now in charge of the Justice Departments Russia investigation. Its Washington correspondent, John Gizzi, regularly gets called upon to ask questions during White House press briefings.
Trump has made it clear that he is frustrated by the investigation into Russian interference in the presidential election, which USA intelligence officials have said was done to tilt the outcome in Trump's favor.
"I think it's pretty clear by what one of his lawyers said on television recently".
Deputy Attorney General Rod Rosenstein says he will "defend the integrity" of the special counsel's investigation into Russian interference in the US elections.
In an April 2015 interview with The Palm Beach Post, Ruddy said he liked then-GOP frontrunner Jeb Bush for president.
And deputy White House press secretary Sarah Huckabee Sanders said simply: "Chris speaks for himself".
Journalists and experts have also noted that Trump's close aide Newt Gingrich has also publicly denounced Mueller after lauding him when he was initially chosen as the special counsel. "If there were not good cause it wouldn't matter what anyone said".More news: What happens to Instacart in the Amazon-Whole Foods deal?
Conservative commentator Ann Coulter offered a similar message, tweeting, "Now that we know TRUMP IS NOT UNDER INVESTIGATION, Sessions should take it back & fire Mueller".
There is no reason to fire the independent head of the FBI investigation into possible collusion between Russian Federation and the Trump campaign, the Justice Department's number two said Tuesday.
"The Attorney General may remove a Special Counsel for misconduct, dereliction of duty, incapacity, conflict of interest, or for other good cause, including violation of Departmental policies".
White House staffers have nonetheless sought to talk Trump down from the idea.
Sen. Susan Collins, R-Maine, asked Rosenstein at a budget hearing Tuesday what he would do if Trump ordered him to fire Mueller, who is investigating Russian interference in the USA elections and possible Russian ties to Trump's campaign.
Mueller and former Federal Bureau of Investigation director James Comey worked closely for years in the George W. Bush administration, when Mueller was Federal Bureau of Investigation director and Comey was deputy attorney general. Graham also asked him if it would be a disqualifier to have worked for Hillary Clinton, who ran against President Donald Trump in the election and was a subject of a separate Justice Department investigation into her email practices. And asked about Trumps own contention that he fired Comey with the Russian Federation probe in mind, and regardless of any recommendation from anyone else, Sessions said: I guess Ill just have to let his words speak for themselves.
As attorney general, Sessions is unlikely to answer in detail questions about conversations he's had with Trump.
He said he would agree to dismiss Mueller only if there were a legitimate basis to do so, and an order from the president would not necessarily qualify.