The injury occurred during the Sixers' 127-111 victory over the Atlanta Hawks in Game 3 on Friday night. The veteran will miss the rest of the Eastern Conference semifinal series but could return in the next round if Philadelphia advances.
Green left with 8 minutes, 15 seconds remaining in the first quarter with Philadelphia up 6-4 and was replaced by Matisse Thybulle.
He returned to the bench in the second half wearing a boot on his right leg.
The three-time NBA champion averaged 4.5 points through the first two games of the second round, shooting just 1-for-9 from 3, but dished a career-high eight assists in Philadelphia's 118-102 win in Game 2 to help even the series.
"The next guy has to step up. Not sure who that is going to be yet," coach Doc Rivers said after the win Friday.
Korkmaz started the second half in Green's place. The Sixers are 32-7 this season with a starting lineup of Green, Joel Embiid, Ben Simmons, Tobias Harris and Seth Curry, according to data compiled by ESPN Stats & Information research.
"I don't know the situation about Danny right now, but it doesn't matter who is in, who is out, we just need to step up. I just need to step up if I need to and I will do it," Korkmaz, who started 11 games during the regular season, said Friday. "I am ready. I hope it's not serious. I hope we're going to get everybody healthy. But if he's out, I'm definitely ready to step up."
The Sixers lead the series 2-1. Game 4 is Monday in Atlanta.
ESPN Staff Writer Dave McMenamin contributed to this report.