Alvarez (elbow) signed a minor-league contract with the Tigers on Friday.
Alvarez underwent Tommy John surgery in September and is expected to miss the entirety of the 2023 season. The left-hander will conduct his rehab under the supervision of the Tigers' staff, and he could remain in the organization in 2024 and challenge for a spot in the big-league bullpen. He appeared in 21 games for the Giants in the 2022 season, logging a 5.28 ERA and 1.70 WHIP in 15.1 innings.