The White House, however, said Trump and Ruddy have never discussed the issue.
In 1973, amid the Watergate scandal, President Richard Nixon ordered Attorney General Elliot Richardson to fire special prosecutor Archibald Cox.
Congressional Republicans stood behind Mueller on Tuesday and said he should be given the freedom to conduct his investigation.More news: Sphero's Artificially Intelligent SPIDER-MAN Toy Looks Pretty Spectacular
Technically, it's up to the attorney general to decide what to do with the special counsel. Shortly after Mueller's appointment was announced last month, Trump floated the idea of firing him, but staffers sought to squash the idea, which they believed would "turn a bad situation into a catastrophe", The Times reported Tuesday.
Top House GOP leaders appeared frustrated at the news conference about the multiple questions about Mueller's probe and what might happen if he were forced out of his position.
Trump interviewed Mueller for the director's position days before Rosenstein picked Mueller as special counsel for the Russian Federation probe, Sanders confirmed Tuesday. Comey last week told a Senate panel that Trump had lied about the Federal Bureau of Investigation, and he said he believes he was directed to end a separate probe into former national security adviser Michael Flynn. Trump had also interviewed Mueller for the role of FBI director before Mueller was appointed special counsel in charge of the Russian Federation investigation.
"I appointed him. I stand by it. and I am going to defend the integrity of that investigation", he said.
He told senators during a committee hearing that there would have to be "good cause" for Mueller to be removed from his role.
President Donald Trump speaks on the South Lawn of the White House in Washington, Monday, June 12, 2017, during a ceremony honoring the 2016 NCAA Football National Champions Clemson University Tigers.
Donald Trump has called the ongoing investigation a "witch hunt", a "hoax" and an excuse by Democrats for why they lost the presidential race.
Another distraction for Trump came from the attorneys general of Maryland and the District of Columbia, who filed a lawsuit against him.