Durant, who leads the team with 49 tackles, was injured in the second half of Monday's overtime loss to the Washington Redskins.
Veteran linebacker Tim Dobbins, 31, was signed to replace Durant on the 53-man roster.
Also, starting middle linebacker Rolando McClain missed Wednesday's practice for an undisclosed reason. McClain has missed practice this year for legal and health reasons.
Suspended defensive lineman Josh Brent attended practice Wednesday and did some individual drills on the side. Brent is eligible to start practicing with the team next week.
Quarterback Tony Romo (back), left guard Ronald Leary (groin) and right tackle Doug Free (foot) missed practice. With the Cowboys having a short week, the team's Wednesday practice was more of a walk-through as players didn't wear helmets during the session.