Senate Republicans at a closed-door lunch Thursday discussed how to solve an interparty conflict that has stalled the GOP's top priority: judicial nominations.

Outgoing Sen. Jeff Flake Jeffrey (Jeff) Lane FlakeHow fast population growth made Arizona a swing state Jeff Flake: Republicans 'should hold the same position' on SCOTUS vacancy as 2016 Republican former Michigan governor says he's voting for Biden MORE (R-Ariz.) is opposing judicial nominations until there is a vote on legislation that would protect special counsel Robert Mueller Robert (Bob) MuellerCNN's Toobin warns McCabe is in 'perilous condition' with emboldened Trump CNN anchor rips Trump over Stone while evoking Clinton-Lynch tarmac meeting The Hill's 12:30 Report: New Hampshire fallout MORE from being fired, and give him legal recourse if he is.

"We're trying to figure out how to deal with that. No decision made, but that was a subject of discussion at lunch to try to figure out how to break that impasse and, like I said, no resolution yet," Cornyn said, asked if the special counsel legislation could get a vote in light of Flake's maneuvers.

One potential option floated during the lunch, according to Cornyn, would be to take a nonbinding "sense of the Senate" vote that would put lawmakers on the record.

"Somebody mentioned a possible sense of the Senate resolution, since there were objections raised on the constitutionality of such a bill and objections raised on the improbability that the House would ever pass it and the president would ever sign it," Cornyn said, referring to the special counsel legislation.

Several Republican senators have backed holding a vote on the Mueller legislation, but so far Flake hasn't won over a second Republican to help him block judicial nominees on the Senate floor.

"I'm continuing to work with leadership and make the case for bringing it to the floor, which I think only gets stronger with each passing day. And I also think that we can continue to try to bring it to the floor," she said.

The Senate Judiciary Committee passed legislation to protect Mueller, or any other special counsel, in the event he is fired, but the bill has stalled amid opposition from GOP leadership.