Cobb has not pitched in a game this season. Forearm pain led him to leave a spring training start in March, and an MRI this week revealed the tear.
Instead, Cobb will undergo the surgery, which typically requires at least 12 months of recovery time.
Cobb, 27, is making $4 million in his first year of arbitration eligibility.
Over 166 1/3 innings last season, he recorded a 2.87 ERA (sixth in the American League) with 149 strikeouts and a 10-9 record.
The Rays, who rank fifth in MLB in starters ERA, currently sit in second place in the AL East with a 15-14 record.