Green, who had the operation at the Cleveland Clinic, will miss the remainder of this season.
Doc Rivers said Green's surgery "went well" and the coach hoped the next announcement would be "that he'll be able to play again."
"That would be terrific," Rivers said. "I don't care if it's with us. I want it to be with us, obviously. I just want him to be able to fulfill his life. Really, at the end of the day, that's what we want."
Boston picked up Green from Oklahoma City last February in the trade that send Kendrick Perkins to the Thunder. He averaged 9.8 points and 3.3 rebounds in 26 games for the Celtics. For his four-year career, he has averaged 13.9 points and 5.5 rebounds.