The reigning Heisman Trophy winner was suspended from Florida State's game against Clemson on Saturday after after many students reported he shouted an obscene sexual phrase in the student union.
"As of now, I wouldn't," Dungy said. "I'd have to do a lot of research and go in and talk to him. But you don't want the face of your franchise and the guy you're going to take with one of the top five picks to have all those off-the-field issues. You just don't."
Last November, Winston was accused of sexually assaulting a woman in December 2012. He was never charged, and Florida State has opened a Title IX inquiry into the handling of the school's initial investigation of the allegations.
In April, the quarterback was cited for shoplifting $32.72 worth of crab legs and crawfish from a Publix.
Dungy said if Winston were to fall to the late first round or early second it could be a little easier to decide to select him, but warned that the risk factor is still there.
"If he becomes your starting quarterback and the issues aren't resolved, you still have the same problems. You have the same thing with Johnny Manziel whether you took him with the first pick or the 25th pick."
FSU coach Jimbo Fisher said Sunday that Winston will return as starter for the team's game against NC State on Saturday.
- Molly Geary