Chapman is set for surgery Monday after getting another opinion on his right hip tendinitis.
The 27-year-old Chapman, a back-to-back Gold Glove winner at third and an All-Star in 2019, was hitting .232 with 10 home runs and 25 RBIs this season.
"Plays about as hard as anybody in the game. Unfortunately, he won't be with us," manager Bob Melvin said. "This one stings some, but we have to rally around it, and play with his spirit. We know he's pulling hard for us. You just have to get through it."
Puk is set to undergo surgery Wednesday on his pitching shoulder. He hasn't pitched this season because of inflammation in the shoulder that required injections. He underwent Tommy John surgery on his left elbow in 2018.
"For A.J., obviously, it's been a tough road for him," Melvin told reporters. "Back and forth, ramping up, having to shut down. Obviously there's something going on that he needs to take care of."
The 25-year-old Puk had been set to join the Oakland rotation in 2020 after making 10 appearances in 2019, going 2-0 with a 3.18 ERA in 11⅓ innings.
The Associated Press contributed to this report.