A source told the newspaper that the injury will sideline Davis indefinitely, and team president of basketball operations Danny Ainge alluded to that possibility, telling the Globe, "It looks like he's going to be out for a while."
When asked about the circumstances of the injury, Ainge said he "would not confirm or deny" that it occurred in an off-court altercation, the Globe reported.
"I am not making a public statement until we have all of the facts relative to the injury," Ainge said, according to the report. "We sent him for a scan and further tests."
Celtics coach Doc Rivers, who spoke to the media after the Celtics practiced Monday, said the injury occurred before practice, but "other than that, no news. Don't know anything."
Rivers said he also did not know which hand was injured. "Honestly, we just sent him out and we'll find [out] more about it probably later," he said.
Davis, who re-signed with the Celtics in the offseason, is heading into his third season in Boston. He averaged 7.0 points and 4.0 rebounds a game off the bench last season, but those totals spiked to 15.8 points and 5.6 rebounds per game in the playoffs, when he was a starter filling in for the injured Kevin Garnett.
The Celtics open their season Tuesday night in Cleveland.