The 33-year-old, who joined the Championship outfit in the summer, had been troubled with the injury in recent weeks.
Royals boss Nigel Adkins told the club's official website: "It's a big blow. He's a talented, talented player. He's been really good, we were really excited to get him in the summer.
"We've been managing him for the past few weeks but he'll be out for a few more weeks now, we just needed to tidy things up a bit."
Bridge started his career at Southampton before joining Chelsea in 2003.