Lee was placed on injured reserve earlier this month with what was described as a pelvic strain, per Todd Archer of ESPN. He will be eligible to return after three weeks, but the timeline will now keep him out until at least mid-October.
The 34-year-old appeared in all 16 games last season for the first time in his 10-year career.
Lee finished last season with 86 tackles to go with one sack and one interception and has mostly been an impact player during his career when healthy. He earned two Pro Bowl selections and was a first-team All-Pro in 2016.
However, his career has been sidelined by a torn ACL, concussions, hamstring issues and more.
The injury will be especially damaging for the Cowboys, which will also be without Leighton Vander Esch for six to eight weeks as a result of a broken collarbone suffered in Week 1, per Archer.
Dallas still has a Pro Bowl linebacker anchoring the middle of the defense in Jaylon Smith, but the lack of depth could be an issue at the position for the next month and beyond.
Joe Thomas is expected to take on a bigger role at linebacker along with Luke Gifford.