"While the president has the right to, he has no intention to do so", deputy White House press secretary Sarah Sanders told reporters on Tuesday when asked if Trump meant to fire Mueller.
For now, the staff has prevailed.
On Tuesday night White House spokeswoman Sarah Huckabee Sanders told reporters flying with Trump on Air Force One that the president wasn't going to fire Mueller.
Last week, Comey told the Senate Intelligence Committee that he had informed Trump that there was no investigation of the president's personal conduct, at least while he was leading the FBI.
Trump, according to The Times, has brought up the legal and political implications of firing Mueller, whom he reportedly thinks is "incapable of an impartial investigation".
Terminating Mueller would be the second time Trump has fired a senior law enforcement official investigating the Russian Federation issue - the first being Federal Bureau of Investigation director James Comey - and would run the risk of sending an even more unambiguous signal the president is axing everyone closing in on a truth he would like to keep secret.
White House press secretary Sean Spicer denied Ruddy's remarks, but his comments fueled most of the media narrative on Wednesday.
Rosenstein is charged with Mueller's fate because Attorney General Jeff Sessions recused himself from all matters having to do with the Trump-Russia investigation.
Former Speaker of the House Newt Gingrich said Tuesday on ABC's "Good Morning America" that while he feels "independent counsels are very risky", he doesn't think Trump should force Mueller out. Then, on Monday evening, Newsmax CEO and Trump pal Chris Ruddy said that the president was "considering perhaps terminating the special counsel", which, Ruddy added, "would be a very significant mistake".
Several senior Trump aides believe that Comey went public with his doubts about the president's behavior and trustworthiness with the intention of steering Rosenstein toward appointing his friend Mueller, according to one longtime Trump associate who remains close to the White House.
Comey's carefully worded comments, and those of Andrew McCabe, who took over as acting Federal Bureau of Investigation director, suggested to some officials that a probe of Trump for attempted obstruction may have been launched after Comey's departure, particularly in light of Trump's alleged statements regarding Flynn.
Mueller, who was appointed as special counsel to oversee the FBI's Russian Federation investigation last month, was roundly praised by Democrats and Republicans as the right person for the job.
Starr on Monday called Mueller "a great man" who is "honest as the day is long" and has a "great, great team of complete professionals".
Under the present circumstances, however, and not without regret, I believe that the same ethics rules the attorney general cited in his Tuesday testimony counsel against Mueller's continuing to serve in the role assigned to him.
Jared Kushner, the president's son-in-law and adviser, supported firing Comey, but he has been less pugnacious lately, administration officials said.
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