Shams Charania of The Vertical initially reported the news.
He suffered the injury during Cleveland's 124-107 loss to the Indiana Pacers on Wednesday night after leaving the game in the second quarter.
Thompson, 26, is averaging 4.4 points and 6.1 rebounds in 21.6 minutes per game this season.
Thompson's role for the 3-5 Cavaliers has been somewhat in flux this season. After starting the year on the bench while the team experimented with Kevin Love at center, Thompson has moved back into the starting lineup, earning five starts.
The Cavs have been somewhat in flux as well, adjusting to a largely overturned roster. The team's early struggles can be easily attributed to the defensive side of the ball, where they are giving up 111.9 points per game, 26th in the NBA.
Thompson's injury won't help the team's efforts to reverse course. In his absence, Jae Crowder is likely to return to the starting lineup, bumping Love back to the 5, while Ante Zizic and Channing Frye could see their minutes increase. In general, however, the Cavaliers are likely to play with smaller lineups in Thompson's absence as they look to iron out their chemistry and defensive woes over the next month.