The Rams cut him on Tuesday after they agreed to send their fourth and sixth round picks to the Buccaneers in return for S Mark Barron.
Keenum, 26, had one year remaining on his contract and was set to earn $495,000 in the current season.
Keenum has not yet been active this year, which could hurt his chances of being picked up by another team. Still, the NCAA’s all-time leading passer probably showed enough play-making ability in 2013 to get a shot elsewhere.
Last year, Keenum appeared in eight games and completed 54.2 percent of his passes for 1,760 yards, nine touchdowns and six interceptions.