The two-time Gold Glove winner has been playing with the injury, which he sustained when he got crossed up on a pitch by reliever James Karinchak in Chicago on April 13.
Pérez will visit hand specialist Dr. Thomas Graham in Dayton on Thursday to determine the best course of action.
Pérez’s batting average has plummeted from .238 to .131 since he got hurt.
Austin Hedges will likely handle the bulk of the catching duties while Pérez is out. The Indians have recalled catcher René Rivera from Triple-A Columbus.