DNC Communications Director Brad Woodhouse said Kaine, a former Virginia governor, told his law school class he was leaning toward seeking the Democratic nomination in the race to succeed retiring Sen. Jim Webb, D-Va., but hasn't made a final decision, The Hill reported.
"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.
Kaine won't make or announce a final decision until he has completed a "final round of consultations with folks about how he can best serve the president, the people and the causes he cares about," Woodhouse said.
He also wants to be assured the Democratic Party would be in "good hands" and that he has the necessary support to mount a successful campaign, the statement said.
If Kaine decides to run, it could set up a high-voltage tilt with former Sen. George Allen, the Republican who is looking to reclaim the seat he lost to Webb.
Notable deaths of 2014 [PHOTOS]