Webb was the Cowboys fourth-round pick (No. 114 overall) in the 2013 NFL draft. Webb finished 2013 with 13 tackles in 15 games.
Webb injured his hip flexor and would have missed the Cowboys' final two preseason games.
ESPN notes that the Cowboys were down to one running back -- DJ Adams -- for Thursday night's preseason finale against the Denver Broncos due to injuries to Joseph Randle (oblique) and Ryan Williams (cramping).
The Cowboys do not want DeMarco Murray or Lance Dunbar to play.
Tanner played for Dallas from 2011-13. He was cut by the Indianapolis Colts during the preseason. In 39 games for the Cowboys, Tanner has 56 carries for 149 yards.
- Sarah Barshop