Multiple outlets have reported that Mueller's investigators want to talk to the president in particular about his decision to fire former FBI Director James Comey in May 2017, and to fire former national security adviser General Michael Flynn in February that year.
USA media reported he gave the memos to the Mueller inquiry, and that they supported former FBI director James Comey's account of the circumstances of his dismissal last May, which initially led to the Mueller investigation.
King said that it would be a "crisis" if Trump fired Mueller, but noted that he does not see it as an impeachable offense.
Republican hardliner Senator John McCain (R-Arizona) also threw his weight behind Mueller, saying it is "critical" that Mueller is allowed to proceed with the probe "unimpeded". "I am confident he will be able to do that", he said. "We have a system based upon the rule of law in this country, we have a justice system, and no one is above that justice system".
While Trump's recent Twitter meltdown has raised speculation that he may fire the special counsel, some lawmakers have cautioned against it. For the most part, however, the G.O.P. caucus stayed quiet-a tacit stamp of approval, perhaps, that shuttering Mueller's probe could be justified.
Graham, a co-author of legislation that would make it harder for a president to fire a special counsel, said it was very important that Mueller be allowed to proceed without interference and that many Republicans share this view.
"I don't see the president firing him", the Oklahoma City Republican told ABC's "This Week" on Sunday.
Trump has repeatedly attacked McCabe since last summer, when he became acting Federal Bureau of Investigation director following James Comey's firing. Trump can not directly dismiss Mueller and would have to direct Rosenstein to fire the special counsel - a scenario GOP lawmakers say won't happen. Any dismissal, for cause, would have to be carried out by Deputy Attorney General Rod Rosenstein, who appointed the counsel and has continued to express support.
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Sky News Defence Correspondent Alistair Bunkall said it is believed the aircraft was a two person plane. At RAF Valley on the island of Anglesey, wreckage from the jet could be seen close to the runway.
Others conveyed moderate concern over the president's tirade against Mueller and the special counsel. "That is unacceptable in a democracy", Senator Durbin said. For now, they predicted, Trump will snipe at Mueller from the outside.
Graham called for Mueller to be able to carry out his investigation into Russian interference in the 2016 election "independent of any political influence".
Does anyone think this is fair?
Later, a reporter asked what was "behind" Trump's tweets attacking Mueller.
Mr Trump also challenged the honesty of Andrew McCabe, the newly fired FBI deputy director, and James Comey, the bureau's former director whom the president fired past year over the Russian Federation probe. Comey could have known that McCabe was interacting with the reporter, without authorizing him to be an anonymous source.
Aides and friends say they understand Trump's frustration.
Those memos could be fodder for Mr Mueller's probe regarding potential obstruction of justice by the United States leader.
Marc Short, Trump's congressional liaison, said the frustration is "well-warranted" because "there has been no evidence whatsoever of collusion". He brought up a hypothetical situation where a president might decide to evade the Secret Service and go for a drive, possibly drunk, only to strike and kill a pedestrian.