The Cavaliers didn't provide a timetable for Gibson's his return or recovery.
The 26-year-old originally hurt the ankle during a March 19 game against the New Jersey Nets. X-rays didn't reveal any broken bones, but the ankle failed to respond to subsequent treatment.
Gibson then underwent an magnetic resonance imaging exam at the Cleveland Clinic's Center for Sports Health, during which the tendon damage was revealed, the team said.
The six-year NBA veteran is averaging 7.5 points, 2.9 rebounds and 2.2 assists in 35 games this season.