However, the 31-year-old, who has not missed a game since entering the NFL in 2005, believes that he will be back soon enough, although he acknowledges that next Sunday's trip to the Philadelphia Eagles may come too early for him.
"My body recovers really fast. I'm not going to be out that long," Ware told reporters.
"It wouldn't be 100% if I had to go out and play right now. There wouldn't be a point to this day. If I had to play, I wouldn't. I couldn't get off the ball."
The Cowboys are currently level with the Eagles at the top of the NFC East, meaning that the winner of Sunday's match would take sole control of the division.