As a former Trump campaign surrogate, Sessions insisted he kept to his recusal from the Russian Federation probe even while he had a hand in firing James Comey as FBI director - which Trump later justified based on the Russian Federation investigation.
"I think he's weighing that option", Ruddy said in an interview Monday with Judy Woodruff of "PBS NewsHour".
And he essentially warned Trump that if Mueller were to be fired, Rosenstein wouldn't be the one to do it, saying he saw no "good cause" to consider such a move.
The former director said that after an uncomfortable one-on-one Oval Office meeting in which Mr. Trump urged him to drop an investigation into former National Security Adviser Michael Flynn's contacts with the Russians and other foreign actors, he asked that Mr. Sessions not leave him alone with the president again.
Rubio got Sessions to confirm details of that meeting, although the attorney general and Trump ally was not in the room and therefore could not corroborate whether Trump or Comey is telling the truth about the details of the conversation.
Gingrich and Ruddy said it would be a "mistake" for Trump to remove Mueller, although both said they see the probe as unnecessary.
"He didn't recall this, but I responded to his comment by agreeing that the Federal Bureau of Investigation and Department of Justice needed to be careful to follow department policy regarding appropriate contacts with the White House", Sessions said.
After Trump fired James Comey in an attempt to quash the Russian Federation scandal investigation, he warned the former FBI director that he'd "better hope that there are no "tapes" of our conversations before he starts leaking to the press!"
The comments come amid increasing frustration at the White House and among Trump supporters that the investigation will overshadow the president's agenda for months to come - a prospect that has Democrats salivating. Comey's decision to announce a year ago that Clinton would not be prosecuted over her emails was a "usurpation" of the Justice Department's authority, Sessions said.
Tillerson warns against steps that cut off talks with Russia
Tillerson had argued unsuccessfully that the United States should remain a part of the agreement, to maximize USA leverage. A day earlier, Tillerson told senators at a meeting of the Senate Foreign Relations Committee that the policy had changed.
Top Justice Department officials testified on June 13 that it's actually up to Deputy Attorney General Rod Rosenstein, who originally appointed Mueller to his post, to decide whether he should go, though the president could bypass that protocol.
Sessions said during questioning by senators that allegations of Trump-Russia collusion outlined in a private intelligence dossier produced by a former British intelligence officer Christopher Steele were false.
"I can not and will not violate my duty to protect the confidential communications I have with the president", he said. Even though it is reported that the president understands how risky firing Mueller will be, those close to him feel that Trump is "so volatile they can not be sure that he won't change his mind if he finds out anything to lead him to believe the investigation has been compromised".
That may have been by design, according to a person who spoke to Trump on Tuesday.
LESTER HOLT: And while all of that was playing out today, the future of Robert Mueller, the special counsel in the Russian investigation has called into question.
Schiff told MSNBC's "Morning Joe" that "I don't think the Congress would sit still and allow the president to pick his own investigator".
Sessions recused himself from the Russia probe in March after it was revealed he twice met with the Russian ambassador during the campaign but failed to say so at his confirmation hearing.
He denied that he violated his recusal by advising the president to fire the then-FBI director, James Comey.