Tim Kaine, former Virginia governor and chairman of the Democratic National Committee, is increasingly likely to run for U.S. Senate next year, his spokesman said Monday.

“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; he is assured that the Democratic Party will be in good hands should he choose to make the race and leave the DNC,’’ spokesman Brad Woodhouse said. “He has the support that would be necessary to mount a successful campaign and he completes commitments for travel and fundraising he has made to the party and the president through at least the end of the month.”

Kaine, who teaches at the University of Richmond law school Monday mornings, told his class Monday that he was likely to run. The students told members of the media, setting off renewed speculation.

Kaine had already consulted with President Obama and been encouraged to run by U.S. Senate Majority Leader Harry Reid, both of Virginia’s senators and countless state officials. On Feb. 19 at Virginia Democrats’ Jefferson Jackson Dinner, he was greeted by enthusiastic crowds of activists in his home town and by those wanting him to run.

Several other Democrats have been mentioned as possible contenders, but all are waiting for a decision by Kaine. Defeated representative Tom Perriello is the most prominent name. Other possible candidates include former representative Rick Boucher, state Sens. Chapman “Chap” Petersen (Fairfax) and A. Donald McEachin (Richmond) and state Del. David L. Englin (Alexandria).

Former senator George Allen, who lost the seat in 2006 to Webb, and tea party leader Jamie Radtke are running for the GOP nomination. Several other Republicans are considering a run.