Diggs suffered the injury during last Saturday's win at Arkansas.
The junior and brother of Minnesota Vikings wideout Stefon Diggs has 20 tackles, six pass breakups and an interception in six games this season.
With Diggs out, Alabama will likely turn to true freshman Patrick Surtain II as his replacement
"We're obviously very thin and need some guys to step up and develop," Saban said. "But Patrick has played quite a bit at his position. We do have two guys that have some experience, but we don't have a lot of depth right now.
Alabama hosts Missouri on Saturday before going on the road to Tennessee. The Tide will then have an open week.