Boone said Green, who was placed on the injured list Saturday because of his ailing right elbow, is still deciding on who will perform the surgery, which will end his season.
"It's tough news, but with this surgery, it's usually something that pitchers are able to return from," Boone told reporters, according to MLB.com. "I know he will. He's an outstanding athlete that takes great care of himself and has a really good work ethic. I know that in time, he'll be fine. I expect him to pitch several more years at the level we've come to expect."
The right-hander was injured during Thursday's loss in Baltimore. This season, Green is 1-1 with a 3.00 ERA in 14 appearances.
Since becoming one of New York's high-leverage relievers in 2017, he has a 2.96 ERA in 260 appearances. Last season, Green was 10-7 with a 3.12 ERA in 67 appearances, striking out 99 hitters in 83⅔ innings.
The Associated Press contributed to this report.