The 12-year veteran will undergo an MRI on his calf on Saturday, sources told ESPN.
"It's a calf injury. Just my doctor's degree, I would tell you that calf injuries aren't great," Sixers coach Doc Rivers said after the game.
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," Rivers said. "I'm not ruling Danny out, but I'm pretty much ruling him out. I doubt if he plays the next game."
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.
"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," said Korkmaz, who started 11 games during the regular season. "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 took a 2-1 series lead. Game 4 is Monday in Atlanta.