Under Justice Department rules, Mr Trump would have to order Deputy Attorney-General Rod Rosenstein to rescind department regulations protecting a special counsel from being fired for no good reason, and then to fire Mr Mueller.
Member of the House Intelligence Committee Adam Schiff, a California Democrat, made it clear the House would not stand down if the president fired Mueller.
Some have alleged that Mueller can not be a fair special counsel because of his relationship with former FBI Director James Comey, who was sacked by President Donald Trump last month.
"Director Mueller is going to have the full degree of independence that he needs to conduct that investigation appropriately", Rosenstein said. A senior White House official confirmed that the President had interviewed Mr Mueller for the FBI post in the Oval Office the day before Mr Rosenstein tapped him to be the special counsel in the Russian Federation investigation.
On Monday evening, Chris Ruddy, CEO of the right wing news website Newsmax and loyal friend of the president told PBS News Hour he believed Trump was weighing the option of firing Mueller. But I'm noticing a trend here: If something is bad for Republicans, Newt thinks it should be abolished.More news: Theresa May offers support package for London fire victims
"You're creating a debate that's not happening", said House Majority Leader Kevin McCarthy on Tuesday morning, responding to a reporter's question about the rumor.
Deputy Attorney General Rod Rosentein offered assurances to lawmakers on Tuesday that he is exclusively responsible for Mueller's fate since Attorney General Jeff Sessions has recused himself from the Russian Federation investigation.
And the Daily Caller reported on Tuesday that included an interview with Mueller.
The conservative commentator Ann Coulter offered a similar message, tweeting: "Now that we know Trump is not under investigation, Sessions should take it back and fire Mueller".
Since being appointed special counsel in May, Mueller has built a team of formidable legal minds who've worked on everything from Watergate to Enron.More news: 58 people presumed dead after Grenfell Tower disaster
While Ryan did not say what he and other GOP lawmakers would do if Trump sought to fire Mueller, his words signaled that such a move would not be tolerated and may have been meant to influence the president. He added, "Don't waste time trying to undermine one of your few allies".
This article has been update to include a comment from Deputy Attorney General Rod Rosenstein.
Spicer had said that Ruddy was at the White House Monday to meet with White House aides, but did not speak with the president.
Schiff says that if Trump fires Mueller, Congress might name its own independent counsel to investigate the case. "He had a private conversation with the President on his views about all sorts of matters, potentially about the investigation, I don't know". Ruddy is definitely on Trump's side, reminding viewers on CNN that "there is no evidence of wrongdoing, there's no evidence of collusion, there's no evidence of obstruction".More news: Defending champion Johnson out of US Open
(Trump has "no intention" of firing Mueller, White House spokeswoman Sarah Huckabee Sanders said last night).
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