Medel, an £11million capture from Seville last summer, was replaced at half-time of Saturday's 2-1 FA Cup fifth-round defeat at home to Wigan.
The 26-year-old Chile international later posted pictures of himself on social networking site Instagram undergoing a scan.
Medel added a post which read: "Well, this is football. A tear, I am out for several weeks. (I hope) to recover as quickly as possible."
Medel's absence comes at a crucial point in the season as Cardiff aim to avoid relegation from the Barclays Premier League.
Manager Ole Gunnar Solskjaer's side are currently one place and two points off the bottom of the table, with 12 matches remaining.