Pinillos, who joined Forest in the summer of 2015, picked up the injury during the Championship game against MK Dons at the City Ground on Saturday afternoon.
The 23-year-old left-back was substituted in the 71st minute, as the Reds went on to win the encounter 2-1.
The Spaniard has taken to social media to thank the Forest fans for their support and has assured them that he will come back stronger.
Thankyou all for the messages of support , but I think I'll be away from the pitch for a while. On Monday I'll know more of the injury,‼️.— Dani Pinillos (@DaniPinillos3) December 19, 2015
Forest are 15th in the Championship table at the moment with 28 points from 22 matches.
Dougie Freedman’s side are six points behind Ipswich Town, who are currently sixth in the standings and occupy the final playoff spot.
The Reds will return to league action next Sunday when they take on Leeds United at the City Ground.