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(CNN) —  

Republicans are bluntly warning President Donald Trump to lay off his attacks on special counsel Robert Mueller, but there is scant support among the GOP to pass legislation that would prevent Trump from firing the special counsel.

Republican Senate leaders threw cold water Monday on passing a bill to protect Mueller, calling it unnecessary despite Trump’s increasingly scathing attacks against the special counsel and his team.

But the Republican who chairs the Senate Foreign Relations Committee, Bob Corker of Tennessee, told CNN that lawmakers should include legislation to protect Mueller on a must-pass bill to keep the government open past Friday. He warned there would be “total upheaval” in the Senate if Trump were to fire Mueller.

“I can’t possibly imagine why Senate leadership wouldn’t place a protection in this (spending bill) that’s coming through,” Corker told CNN. “That would be the perfect place for them to deal with it.”

Trump tweeted again Monday that the Russia investigation was a “total WITCH HUNT” and he attacked Mueller by name in a tweet for the first time on Saturday. Trump’s lawyer John Dowd, meanwhile, issued a prayer Saturday that Deputy Attorney General Rod Rosenstein would “bring an end” to the Mueller probe.

Republicans have warned there would be severe consequences if Trump moved to fire Mueller following his latest tweets. The most senior Republican in the Senate, Orrin Hatch of Utah, bluntly warned Monday that firing Mueller would be “the stupidest thing” Trump could do.

But Hatch said he didn’t see a need to pass any legislation, a sentiment that’s echoed by Republican leaders.

“I don’t think it’s necessary,” Senate Majority Whip John Cornyn told CNN. “I think it’d be a bad mistake for the President to fire the director. And I don’t think he’ll do it, so I don’t see any benefit in trying to pass a law.”

Cornyn, a Texan who’s the second ranking GOP member of the chamber, noted that any law the Congress passed would then go to the President’s desk.

“Obviously, legislation requires a presidential signature. … I don’t see the necessity of picking that fight right now,” he told reporters.

Senate Majority Leader Mitch McConnell has yet to weigh in on the President’s latest tweets about Mueller. McConnell has long believed he does not need to respond in real time or react to every controversial tweet or other action by Trump, according to people close to the Kentucky Republican.

McConnell’s staff would only refer reporters to a January statement from McConnell when he said he didn’t think Mueller needed any special protection because he didn’t believe the President would fire him. An aide said McConnell’s position had not changed since then.

At least one Republican wanted to see McConnell speak out: Sen. Jeff Flake of Arizona, a vocal Trump critic, told reporters was disappointed by silence from McConnell over Trump’s attacks on Mueller. Flake noted McConnell’s previous reasoning, which is that the White House has said Trump isn’t considering firing him, arguing that the White House said the same about outgoing Secretary of State Rex Tillerson just weeks ago.

There are two bipartisan Senate bills – one authored by Cory Booker, a New Jersey Democrat, and Lindsey Graham, a South Carolina Republican, the other from Chris Coons, a Delaware Democrat, and Thom Tillis, a North Carolina Republican – that would make it more difficult to remove Mueller from his post. But there’s been little movement on the measures following a Senate Judiciary Committee hearing on the matter last fall.

But even the Republican cosponsors of those bills said they didn’t see a need for the legislation at this time.

“I don’t think there’s any imminent need to do it today or this week,” Tillis said, adding he believed the plan amounted to a “good governance” bill that should be eventually passed.

“Not right now,” Graham said. “We have enough problems with the omnibus now. I’ve got zero concern that the President or his team is going to fire Mueller.”

Sen. Susan Collins, a moderate Republican from Maine, also said she didn’t think the President would fire Mueller when she was asked about passing bills to protect the special counsel, telling CNN there were already institutional protections in place.

“I do not think that the President is going to order anyone to fire Mr. Mueller. That would be a terribly serious mistake,” Collins said. “And it’s important to remember that the President cannot directly fire Mr. Mueller, only the deputy attorney general can do that under the department’s regulations. And he can only do it for good cause. There is no good cause to fire Mr. Mueller.”

Cornyn and Collins acknowledged that the President’s tweets were concerning, but they didn’t think it was a sign he was about to take action against Mueller.

“I mean, I wish he’d quit tweeting about it,” Cornyn said, before adding, “but no, they don’t concern me that he’d fire him.”

CNN’s Ashley Killough and Daniella Diaz contributed to this report.