DNC Chairman Tim KaineTimothy (Tim) Michael KaineTrump impeachment trial to begin week of Feb. 8 Democrats float 14th Amendment to bar Trump from office Trump impeachment article being sent to Senate Monday MORE is "increasingly likely" to run for Senate in Virginia but hasn't made a final decision, a top spokesman said Monday. 

Brad Woodhouse, the communications director for the Democratic National Committee (DNC), said that Kaine informed his law school class that he's leaning toward jumping in the Virginia Senate race.

But, Woodhouse stressed, Kaine has made no final decisions and is continuing a "final round of consultations" with President Obama and Democratic Party officials.

"In response to a student’s question, Governor Kaine told his law school class today what is already widely known which is that he is increasingly likely to run," Woodhouse said in a statement.

"However, 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; 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," Woodhouse added.

If Kaine joins the race, it would set up a top-tier matchup with former Sen. George Allen (R), who's looking to reclaim his seat.

Kaine's decision has been closely watched not only for its implications on that race, but also on Obama's reelection, because the president would need to choose a new leader of the DNC heading into his own reelection effort in 2012.