A student told a Virginia news radio station Monday that former governor and current DNC Chairman Tim Kaine told a law class at the University of Richmond that he is running for the U.S. Senate seat being vacated by Sen. Jim Webb, D-Va. DNC spokesman Brad Woodhouse tells Fox News that Kaine has not made a final decision, but that the DNC Chair is "increasingly likely" to run.

"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."

Former Virginia Governor and U.S. Senator George Allen has already announced he is seeking the GOP nomination for the seat and Tea Party Candidate Jamie Radtke will run against Allen in the primary.

If Kaine enters the fray, the race in the Old Dominion becomes one of the most closely watched races of the 2012, pitting a popular former governor and senator against the sitting chairman of the DNC who himself was a popular governor of the state.

Webb, who is serving his first term, announced in February that he won't seek reelection in 2012.