But after reports last week that Trump ordered Mueller fired and White House counsel Don McGahn refused, there is growing support among some Senate Republicans for stepping in to protect Mueller through bipartisan legislation introduced in August.
"Mueller is the best person to look at it", said Graham, describing the allegation as grave if proved true.
"As a matter of fact, I think Mr. Mueller is the ideal guy to get to the bottom of all this, and he will", Graham said.
"And that was seven months ago and the White House counsel is still on the job, and Mr. Mueller is still aggressively investigating, and that's as it should be".
CNN reports Republican Sen. But she didn't offer a timeline.
On NBC's "Meet the Press," he said, "I don't think there's a need for legislation right now to protect Mueller".
There have been conversations about protecting Mueller in the past, but last week's report by The New York Times, claiming that Trump wanted to fire the investigator last summer, brought new attention to the matter, even though Trump denied the report and called it "fake news". He said Trump and his team "have fully cooperated" with the investigation.
Either way, he and Trump's other attorneys encouraged the president to keep Mueller in place.
On Sunday, lawmakers praised Mueller's impartiality and expressed confidence that he would be able to conduct a full, wide-ranging investigation.
Graham said: "It's pretty clear to me everyone in the White House knows it'd be the end of President Trump's presidency if he tried to fire Mr. Mueller".
Democrats immediately seized on the report, saying they would try to ensure that continuing budget negotiations included legislation to protect the special counsel.
That said, Rosenstein might actually be in the same position as Mueller. Instead, he said the President is having a tough time adjusting from his life as a businessman, where he "had total control". "He feels like there's been millions of dollars of taxpayers' dollars spent and no evidence yet of collusion".
But Assistant Attorney General Stephen Boyd - a Trump appointee - said it "would be extraordinarily reckless" for Nunes release the classified four-page memo without giving the DOJ and the Federal Bureau of Investigation a chance to determine if it could compromise national security. Collins said it's "helpful" to know what the White House wants as Congress moves ahead, but suggested legislators would forge their own path in formulating a bill. "And I'll be glad to pass it tomorrow", Graham said on ABC's "This Week with George Stephanopoulos".