Turner, who leads the NBA with 4.2 blocks per game, suffered the injury vs. Portland on Jan. 14. He has met with two hand specialists and will be reevaluated in the next few days, sources told ESPN.
Despite the injury, Pacers coach Nate Bjorkgren said Turner still attended Tuesday's practice in good spirits with his hand wrapped.
"He's a stud. ... There's nothing that's going to stop him," Bjorkgren said following practice. "There was first news that this could be two or three weeks and all that. He comes up to me and he's like, 'Coach, no way.' ... The thing I like most about him is what he said to me today and just his toughness. He didn't sit out one part of practice. They put like a little pad on his hand and they taped it up.
"I mean it, just his overall toughness is very impressive."
Turner's 17 blocks were the most by a Pacers player in his first three games of a season since the stat became official, according to ESPN Stats & Information. The previous high was 14 by Roy Hibbert in 2013-14.