Pomeranz, who started Monday against the Houston Astros, left the game with one out in the fifth inning after experiencing tightness in the shoulder.
A's manager Bob Melvin said the team will determine Pomeranz's status Wednesday and that "there's a chance he could go on the DL."
The 26-year-old is 2-3 with a 4.40 ERA this season.
Fellow A's starter Jarrod Parker underwent surgery to repair a broken right elbow Tuesday. Parker suffered the injury during a rehab start with Triple-A Nashville on May 8. Parker, a 26-year-old who hasn't pitched in the major leagues since Sept. 28, 2013, was on the comeback trail from his second Tommy John reconstructive elbow surgery when the broken elbow occurred.
Because the previously repaired ulnar collateral ligament was not damaged in the break and did not need repair, the surgery is not expected to require as lengthy a recovery as a ligament replacement would.
Information from The Associated Press was used in this report.