Grassley is the chairman of the Senate Judiciary Committee, which is conducting its own separate inquiry into Russia's involvement in the 2016 election.

"I have confidence in Mueller, the president ought to have confidence in Mueller, and I think to answer your question, it would be suicide for the president to want — to talk about firing Mueller," Grassley said Tuesday in an interview on Fox Business.

Sen. Chuck Grassley , R-Iowa, admonished President Donald Trump for his recent remarks entertaining the idea of firing special counsel Robert Mueller , who is investigating potential collusion between the Trump campaign and Russia.

The president tore into the special counsel's investigation Monday evening, calling it a "witch hunt" and "an attack on our country in a true sense."

Asked whether he would fire Mueller, Trump said: "Many people have said, 'You should fire him.' Again, they found nothing."

He also called "disgraceful" reports that the FBI, reportedly acting on a referral from Mueller, searched the office and residence of Trump's longtime personal lawyer, Michael Cohen. Trump's criticism of the raid spilled onto Twitter Tuesday morning.

Trump tweet attorney-client privilege

Grassley said Tuesday that Trump would benefit from keeping quiet on the Mueller investigation.

"The less the president said on this whole thing, the better off he would be," Grassley said.

He also suggested that firing Mueller could play into the Democrats' hands in the 2018 midterm elections.

"I think that maybe Mueller is coming to a dead end, as far as collusion of Trump with Russia in this election — and it looks like a dead end," Grassley said. "Maybe Mueller would appreciate being fired so he would have an excuse for getting out of it and the Democrats would have a good issue in this upcoming election."

Neither the White House nor a spokeswoman for Grassley immediately responded to CNBC's requests for comment.

Another leading Republican senator was more critical of the development in Mueller's probe. In a Fox News interview on Tuesday, Sen. Rand Paul, R-Ky., said he believes Mueller "has abused his authority" in giving the referral to search Cohen's property.

"Going after someone's personal attorney is a great overstep, I think, in the authority of the prosecutor," said Paul, who often espouses libertarian views about privacy protections.

Sen. Richard Burr, R-N.C., told CNBC that the question of whether Congress should protect Mueller through legislation — which Democrats have increasingly called for — is "not even on my radar."

Burr, co-chair of the Senate Intelligence Committee, added that even if Trump was considering firing Mueller, "he'd have to get to Rosenstein first." And as for a Rosenstein replacement, Burr said "nobody would take that job if they knew" that the possibility of firing Mueller was "on the table."

--CNBC's Kayla Tausche contributed to this report.