Senate Majority Leader John Thune, R-S.D., has thrown cold water on the possibility of tweaking filibuster rules to ensure passage of a key Republican election reform bill.

Nevertheless, some Republicans are still calling on Thune to reinstate the “talking filibuster,” where senators who oppose a piece of legislation must hold the floor and speak.

The House-passed SAVE Act – which, among other things, would mandate that all states require photo ID and verify the citizenship status of potential voters – has virtually no chance of garnering the 60 votes needed for it to pass the upper chamber.

Senate Democrats appear universally opposed to the bill, claiming that requiring people to present proof of citizenship, in person, when registering to vote for federal elections will discourage or prevent certain groups of Americans from voting.

Many Republicans, therefore, want to return to the historical method of filibustering, which would allow them to pass the SAVE Act via majority vote once filibustering senators have ceded the floor.

Following Thune’s comments on the issue Wednesday, Sen. Mike Lee, R-Utah, argued that Republicans “can’t afford the consequences of inaction” while Sen. Rick Scott, R-Fla., urged lawmakers to “keep up the pressure.” Sen. Tommy Tuberville, R-Ala., followed up on Thursday as well, asking Thune to do “[w]hatever it takes” to get the bill across the finish line.

The majority leader, however, appears unmoved. Thune had explained to reporters that a talking filibuster could turn into a weeks-long process, and with a partial government shutdown still ongoing, lawmakers do not have the time.

“Once we get on, if we were going to go down that path, it’s very hard to pivot and get back to open up the government,” Thune said. [Reopening the government] is harder to do when you’re in the throes of a talking filibuster.”

Even if Republicans are able to reach a deal with Democrats to reopen DHS and end the shutdown, Thune added, “the talking filibuster issue is one which there is not, certainly, a unified Republican conference, and there would have to be.”

“If you go down that path, you're talking about the need to table what are going to be numerous amendments, and an ability to keep 50 Republicans unified pretty much on every single vote,” he said. “And there isn’t support for doing that at this point.”

Thune promised a vote on the SAVE Act but said the process by which lawmakers take it up is “an open question,” and that the bill being subject to the 60-vote threshold is “a very real possibility.”