Saban said an MRI revealed the injury after the Crimson Tide beat Georgia 35-28 to win the conference title and secure the No. 1 seed in the College Football Playoff. Alabama will face the No. 4 Oklahoma Sooners in the Orange Bowl.
"Hopefully by the time we start practicing again, Tua will be back on the field and ready to go," Saban said.
Tagovailoa exited the game against Georgia during the fourth quarter. The Heisman Trophy front-runner was having arguably his worst performance of the season, throwing just one touchdown and two interceptions.
Jalen Hurts, the former starter turned backup, came on for Tagovailoa and led the come-from-behind win, throwing the game-tying touchdown and running in the go-ahead score.
Hurts, who was 26-2 as a starter and once won SEC Offensive Player of the Year, had been relegated to backup duty all season, appearing in 11 games off the bench.
Hurts actually had a similar injury to Tagovailoa's this year and missed two games in addition to the team's bye week after undergoing surgery.