Birk's agent, Joe Linta, says the six-time Pro Bowl center should be able to return in a few weeks.
The Ravens had agreed to a two-year contract with former Washington Redskins center Casey Rabach on Tuesday to back up Birk, who played in all 16 games for Baltimore last year despite having knee problems.
However, Rabach failed his physical with the Ravens due to a bad shoulder, a source told ESPN.com senior NFL writer John Clayton.
Rabach is recovering from offseason shoulder surgery. According to the source, he should be completely healed in a couple of weeks.
The 35-year-old Birk has not practiced with the team this week, but hopes to be ready for the Sept. 11 opener against Pittsburgh.
Information from The Associated Press contributed to this report.