After initially being praised by Republicans and Democrats for his integrity, Mueller has had his impartiality as special counsel questioned by some Trump supporters in recent days.
Those in the White House have expressed concern that the probe could detract from the president's agenda for months or years to come.
Trump can not directly dismiss Mueller. That leaves deputy attorney general Rod Rosenstein, who appointed Mueller in the first place.
Last Saturday evening, Ann Coulter - whose interpretation of reality is always suspect - fired off an especially loony tweet: "Now that we FINALLY got Comey to admit Trump not under investigation, Sessions should fire Mueller".
The president sacked Comey in early May. "I think there's a conflict there".
"It's a mistake to pretend that this is going to be some neutral investigation", Gingrich said.
The California Democrat said if the president opts to terminate Mueller from his position lawmakers will reappoint Mueller to the position.
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. He told a couple of Russian officials in the Oval Office that the investigation was going to ease up because he fired that "nut bag" Comey.
Both Fleischer and Gingrich had previously vouched for the character and integrity of Mueller, who was appointed Federal Bureau of Investigation director by Bush and whose term was extended an extra two years by Democratic President Barack Obama.More news: Mali forces kill five militants involved in resort attack
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Surely the president knows that firing the special counsel without an absolutely valid, ironclad reason would blow up in his face.
The New York Times reported late Tuesday that Trump did consider firing Mueller, but that his staff talked him out of it, citing people with knowledge of Trump's interactions with staff. Trump discussed the implications of firing Mueller and told those around him that Mueller was part of a "witch hunt", according to the New York Times.
But after the testimony of former FBI Director James Comey last week, Gingrich said he'd changed his mind.
He also declared it a "detestable and appalling lie" to suggest he was aware of or took part in any collusion between Russian Federation and the election campaign that sent Mr Trump to the White House. The person spoke on condition of anonymity to discuss private conversations.
Perhaps Trump will fire Sessions and then pardon him, in order to appoint a new attorney general who would then fire Mueller at the president's command. "I don't think the Chairman will invite the President, a man he campaigned for, because the role of the House Judiciary Committee right now is to protect the President at all costs; shielding the President from tough questions", he said. Still, he added, "I can't imagine that that issue is going to arise".
Ruddy, who did not claim to have spoken to the president about the matter, appeared to confirm comments made by Trump's lawyer Jay Sekulow. Ruddy did not immediately respond to questions seeking clarification.
"At the end of the day, our Founders gave us a system where the president does have the power to take care that the laws be faithfully executed", Katyal said.
Press secretary Sean Spicer told NPR's Tamara Keith, "Mr".