(CNN) Republican Sen. Mike Lee suggested the President pick federal judge Merrick Garland as the next FBI director, a surprising recommendation following Republican efforts to block Garland from a seat on the Supreme Court last year.

"Instead of a special prosecutor, @realDonaldTrump should nominate Merrick Garland to replace James Comey," Lee tweeted Thursday

While he didn't mention it in his tweet, such a move would also open up a highly coveted seat on the DC Circuit, where Garland currently serves as the chief judge.

The Utah senator is a member of the Senate judiciary committee, which refused to hold a hearing on Garland when President Barack Obama tapped him to replace the late Supreme Court Justice Antonin Scalia in early 2016.

Lee pitched the idea to Trump's advisers when he was at the White House on Wednesday, according to an aide. While the reason for his White House visit was unrelated -- it was to discuss tax reform -- he did bring up Garland as a possible FBI director separately, underscoring that the conservative senator is very serious about the idea.

Republicans, who hold a majority in the Senate, kept the seat vacant until a new president was sworn in -- a strategy that proved worthwhile for their party when Trump won the election.

Trump later nominated Neil Gorsuch , who was confirmed in the face of intense anger from Democrats still fuming over the blocking of Garland.

Good idea @SenMikeLee. Former prosecutor Merrick Garland for FBI Director is great idea. Need special prosecutor toohttps://t.co/HjMdwIGxAq — Amy Klobuchar (@amyklobuchar) May 11, 2017

Democrat Amy Klobuchar, another senator on the judiciary committee, responded to Lee's tweet Thursday , saying Garland, a former prosecutor, would be a "great idea" for the FBI. She also called for a special prosecutor to investigate the Trump's campaign's possible ties to Russia.

A few hours later, Klobuchar sought to clarify her tweet, saying she doesn't think Garland getting the gig is realistic.

"To be clear, this isn't going to happen. I RTed bc it's a good idea for Rs to think about consensus FBI candidates," she tweeted

Some Democrats also expressed skepticism about the idea.

"After the way the Senate treated Merrick Garland, I hope this isn't some idea of a consolation prize," Sen. Dick Durbin, D-Illinois, told reporters.

"I doubt that he would consider it," he added. "He's the chief judge of the DC Circuit Court."

Lee's tweet came not long before senators started walking over from their offices to the Senate chamber for a vote on a separate issue. Many had just learned about the tweet or hadn't heard about it when reporters began asking for their reaction.

"I think it's an interesting idea," said Sen. Dianne Feinstein, D-California. "I actually haven't thought about it."

"That's a great suggestion. We'll see where it goes," said Sen. Joe Manchin, D-West Virginia. "That's the first I've heard about it."

"Good choice!" Sen. Lindsey Graham Graham, R-South Carolina, yelled out from an elevator as the doors were closing.

Sen. Steve Daines, R-Montana, described it as an "interesting thought." Asked if he thinks Republicans could get behind it, Daines said he didn't know. "I think he'd be qualified and capable," he added.

Sen. Roy Blunt, R-Missouri, paused for a few seconds before answering and said he liked the idea of creating a vacancy on the DC Circuit. "Maybe I like it better the more I think about it," he told reporters.

Sen. John Cornyn, the No. 2 Republican in the Senate, also perked up when realizing that a seat would open up, which would give Trump an opportunity to appoint a replacement.

"You're getting my attention," he told reporters.

While Sen. Tom Udall, D-New Mexico, said it would be a "concern" to make Garland's seat open, he also said he still hasn't "given up" on the idea of getting Garland on the Supreme Court.

Later Thursday, Lee told reporters he thinks the idea of Garland as the FBI director could probably get votes on "both sides of the aisle." He also praised Garland as someone with a strong law enforcement background and extensive experience as a federal prosecutor.

"I think he fits that bill," he said. "There are other people who would also. But he certainly does."

Asked if he thinks the President will be opened to his idea, he shrugged his shoulders and said: "We'll see."

In the wake of the Comey's firing from the FBI, some senators from both sides of the aisle have been calling for a replacement who can win bipartisan support.

The President's nominee, however, only needs a simple majority of 51 votes to be confirmed in the Senate, which already has a 52-48 Republican majority.