Varejao underwent an MRI on Wednesday after suffering the injury during the third quarter of Tuesday night’s 125-104 win over the Minnesota Timberwolves.
He left Quicken Loans Arena in Cleveland on crutches and with his leg in an air cast.
Cleveland has been aggressive in the past week in trying to acquire another big man, according to ESPN's Brian Windhorst and Marc Stein.
Varejao has missed over 160 games in the past four seasons because of injuries. He signed a three-year, $30 million contract extension in October and averaged 9.8 points and 6.5 rebounds in 24.5 minutes a game for the Cavaliers this season.
Cleveland was seen as a contender to win the NBA championship after acquiring LeBron James and Kevin Love in free agency this offseason. The loss of Varejao will hurt the team's frontcourt depth.
Among the Cavaliers' other frontcourt players are Tristan Thompson and Brendan Haywood.
Cleveland (17-10) faces the Heat in Miami on Christmas Day.
- Scooby Axson and Chris Johnson