Senate Majority Leader Harry Reid could announce his pick to lead the Democratic Senatorial Campaign Committee as early as Tuesday. But he still doesn’t know who it’s going to be.

The Nevada Democrat has been struggling to recruit a DSCC chairman in the wake of the Nov. 2 midterms, and 2012 promises to be a challenging cycle for Senate Democrats. Sen. Michael Bennet (Colo.), who narrowly survived his bid for a full term earlier this month, has apparently ruled out the job after meeting with Reid earlier this week, according to a report in the Washington Post. Sens. Mark Warner (Va.), Al Franken (Minn.) and former DSCC Chairman Charles Schumer (N.Y.) have said they don’t want the job. Current Chairman Bob Menendez (N.J.) is up for re-election next cycle, and he has said he doesn’t want to repeat his tour either.

One senior Senate Democratic source said Reid hopes to name his choice after the leadership elections Tuesday.

Beyond Bennet, other potential candidates for the job include Sens. Mark Begich (D-Alaska,) Tom Udall (D-N.M.) and Mark Udall (D-Colo.), but they also have said they are uninterested. However, another senior Senate source cautioned that many Senators said they don’t want to do it but could ultimately be persuaded. This source said that Reid has not necessarily put the “hard sell” on anyone.

This is not the first time Reid has had to do some convincing to recruit a DSCC chairman. He had to approach Menendez three times before he finally accepted in December 2008. Schumer also needed some wooing to do a second term (he led the DSCC in the 2006 and 2008 cycles), and in both cases Reid offered sweeteners such as influential committee assignments to try to win over his colleagues.

Next cycle, 23 Democrats are up for re-election, and the next DSCC chairman will be tasked with defending incumbents in battleground states such as Missouri and Pennsylvania.

Bennet was appointed in 2009 to the seat previously held by Sen. Ken Salazar (D-Colo.). The former Denver Public Schools superintendent, who had never run for public office until this year, raised nearly $11.5 million and won by a narrow 48 percent to 47 percent.