He said he will fire Mueller only if there was a "good cause" for it, "it wouldn't matter to me what anybody said".
In a series of statements on Twitter Thursday, Trump called special counsel Robert Mueller's investigation into Russian interference in the 2016 campaign a "WITCH HUNT" based on the "phony" premise of possible collusion between Russia and a cadre of Trump campaign associates. So that is what I would do. Lindsey Graham: "Do you know of any reason for cause to fire Mr. Mueller as of this date?"
"I think he's weighing that option", Mr Ruddy said.
Yesterday, a close confidant of Trump, Newsmax website chief executive Chris Ruddy, said following a White House visit that the president is considering firing Mueller. In theory, it would depend on how Congressional Republicans as well as the Department of Justice choose to react to Trump's (hypothetical) move to fire the special prosecutor.
White House press secretary Sean Spicer responded to Ruddy in a statement, saying Ruddy "never spoke to the president regarding this issue".
"The President did talk with him in the days before he was named special counsel", Ruddy told PBS on Monday in the same interview where he said Trump was "considering terminating" Mueller as special counsel.
The authority to appoint a special counsel fell to Rosenstein, after Attorney General Jeff Sessions recused himself from the Russia case because of his failure to disclose during his January confirmation hearing pre-election meetings he had Russian ambassador Sergey Kislyak.
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"Director Mueller is going to have the full independence he needs to conduct that investigation, " Rosenstein said.
Sessions in March stepped aside from the federal investigation into contacts between Russian Federation and the presidential campaign after acknowledging that he had met twice previous year with Kislyak.
Trump's Twitter response followed days of intensifying criticism of Mueller from some conservatives who have charged the former Federal Bureau of Investigation director and his team with political bias and have claimed his relationship to Comey and an earlier meeting with Trump amount to conflicts of interest.
"With respect to this subject, only the president or his attorneys are authorized to comment", he continued.
"Whoever leaked [news of the obstruction investigation] was obviously reading that he was thinking about giving Mueller the boot", an official told the Daily Beast. Still, in light of these new reports, many are likely wondering what would actually happen if Trump tried to fire Mueller. Doubt, however, has been cast on that explanation because of cryptic testimony from former FBI Director James Comey last week.
But people close to Mr. Trump say he is so volatile they can not be sure that he will not change his mind about Mr. Mueller if he finds out anything to lead him to believe the investigation has been compromised.
"I never claimed that I had spoken with the president", Ruddy said. "And the next day he's now maybe using some of that information in his investigation". They did not explain how they would push that idea through a Republican Congress.