A Democratic National Committee spokesman said Monday that DNC Chairman Tim Kaine is “increasingly likely to run” for Senate in Virginia, but added that reports that Kaine announced his campaign to a group of students are premature.

“No final decision will be made or announced until the governor has had a final round of consultations with folks about how he can best serve the president, the people and the causes he cares about,” DNC spokesman Brad Woodhouse said.

Rumors broke out on Twitter earlier Monday that Kaine had announced his decision to run for Senate to students at the University of Richmond.

Woodhouse added that Kaine has been assured that the party will be able to transition smoothly if he opts to run for retiring Sen. Jim Webb’s (D-Va.) seat (The Fix looked at possible replacements in his Monday column), and also that Kaine sees the race as winnable.

“He has the support that would be necessary to mount a successful campaign (as) he completes commitments for travel and fundraising he has made to the party and the president through at least the end of the month,” Woodhouse said.

Meanwhile, one senior party operative told The Fix that the situation is coming together nicely for Democrats. “It looks good,” the operative said.

Kaine has been struggling with the decision since Webb retired a month ago. Initially, he looked unlikely to run, but an extensive effort to get him into the race looks to be paying dividends.

As a former governor, he is the Democrats’ top option in the Commonwealth and would likely have a pretty easy time in the primary.

On the GOP side, former Sen. George Allen (R-Va.) is the early favorite but could face some tea party opposition in the primary.