The school's student newspaper, The Collegian, reports that Kaine, who teaches a leadership studies class at the University's law school, told students on Monday that he had made a decision and would "give it a shot," referring to a run for Senate.
The Democratic Party's communications director, Brad Woodhouse, has since said that Kaine only told students that he was likely to run, and that no final decision had been made.
The paper, however, disputes that claim.
Kaine's name quickly rose to the top of the list of potential Democratic candidates after Jim Webb announced last month he would not be seeking reelection.
A Kaine run could set up a matchup with former Republican Sen. George Allen, who has already announced his candidacy.