Senate Majority Leader Mitch McConnell on Tuesday publicly floated the idea of getting Attorney General Jeff Sessions to take his old Senate seat, as Republicans continue to seek alternatives to Judge Roy Moore.

McConnell made the comments to the Wall Street Journal's CEO Council while talking about who could mount a successful write-in campaign, like Sen. Lisa Murkowski did in Alaska in 2010. McConnell said she was popular in the state and was universally well-known, qualities that Sessions has in Alabama.

"The name being most often discussed may not be available, but the Alabamian who would fit that standard would be the attorney general," McConnell said. "He's totally well-known and is extremely popular in Alabama."

"That obviously would be a big move for him and for the president," McConnell said. He said President Trump is discussing what to do in the Alabama race "in great detail" as he wraps up his Asia trip.

Related: Roy Moore: Mitch McConnell's days as majority leader 'are coming to an end very soon'

McConnell conceded that he isn't sure Sessions would come back after leaving earlier this year to take over as head of the Department of Justice. Some reports were already saying Monday that Sessions had no interest.

"We don't know, but he fits the mold of somebody who might be able to ... in the same way Lisa was able to do it in 2010," McConnell said.

When told by the Wall Street Journal's Gerald Seib that it sounds like he really wants to see Sessions come back to the upper chamber to retake his seat, McConnell said his main priority is to keep the seat in Republican control.

"I'd like to save the seat, and it's a heck of dilemma when you've got a completely unacceptable candidate bearing the label of your party within a month of the election," McConnell said. "It's a very tough situation."

Moore has been dogged by claims of sexual harassment, and many Republicans are calling for him to drop out of the race before people vote on Dec. 12. Moore has not indicated he will quit, and his decision threatens to give the seat to Democrats.

A Democratic win in Alabama would give Republicans a narrow 51-49 majority.