The team added, "He will now undergo a period of treatment and rehabilitation. His status will be updated as appropriate."
ESPN's Adrian Wojnarowski first reported the news.
The Cavs acquired Nance in February as general manager Koby Altman reshuffled the roster amid a disappointing start to the 2017-18 season.
Despite playing with a much better supporting cast than he had with the Los Angeles Lakers, Nance's numbers didn't improve significantly with his move to Cleveland. He averaged 8.6 points and 6.8 rebounds in 42 games in Los Angeles, before putting up 8.9 points and seven rebounds in the second half with the Cavaliers.
Nance was a valuable role player for Cleveland, though. After he arrived, the team's net rating improved by 4.7 points per 100 possessions when he was on the floor, per NBA.com.
Following LeBron James' departure, the Cavs once again find themselves among the NBA's worst teams. At 8-34 they're clearly playing for draft position.
But Nance is one of the few players who's part of Cleveland's long-term future. After signing him to a four-year, $44.8 million extension, the team is reliant on him for the foreseeable future, and losing him for a period of time will stymie their development.