Still, the NRSC job could be good for Rubio if he has ambitions to climb the leadership ladder or seek the presidency, GOP sources said.
"It [will likely get current NRSC Chairman John] Cornyn to the Whip position," one source said, who noted that Minority Leader Mitch McConnell (Ky.)and former Majority Leader Bill Frist (Tenn.) served as NRSC chairmen. "It's not a glamorous job. But what it does do is build a lot of good will and immediate alliances and stronger friendships. ... It's a great spot to build a base of support."
Thune would face an uphill climb for the Whip post. Cornyn is the only declared candidate for the second-ranking leadership post. Current Whip Jon Kyl (Ariz.) is retiring.
Cornyn has been seeking commitments from his colleagues since last year, which gives him a leg up on any challengers, GOP sources said. Plus, Cornyn could benefit from any gains the party makes in the Senate in November, particularly if he helps make McConnell Majority Leader.
Despite being the frontrunner, sources said Cornyn does not expect an uncontested ride into the Whip position. "Cornyn seems to anticipate some challenge, but from whom is an unknown," one GOP operative said.
But with the GOP leadership elections coming within three weeks of the presidential election, there will be little time for anyone to mount a challenge, the operative noted.
If Thune decides to run for a new post, possible candidates for Conference chairman include John Barrasso (Wyo.), who is now Republican Policy Committee chairman - a position considered the proving ground for future Conference chairs. But Roy Blunt (Mo.), who became Republican Conference vice chairman in January, could also throw his hat into the ring.
Sens. Mike Johanns (Neb.) and Ron Johnson (Wis.), who have both sought leadership positions in the past, could also be candidates for Conference vice chairman and policy committee chairman.
Sen. Jeff Flake, R-Ariz., takes a selfie with Faye, a pot belly pig, after a news conference held by Citizens Against Government Waste at the Phoenix Park Hotel to release the 2015 Congressional Pig Book which identifies pork-barrel spending in Congress, May 13, 2015.