Wilson has "structural issues" in his throwing elbow, Manager Bruce Bochy told reporters, adding Wilson would be seeking a second opinion from an orthopedist after the magnetic resonance imaging diagnosis.
If the initial assessment is confirmed, it will likely require season-ending Tommy John surgery -- the second of Wilson's career.
The 30-year-old reliever also underwent ulnar collateral ligament reconstruction in 2003 while at Louisiana State.
Wilson set a Giants' club record with 48 saves as San Francisco claimed a World Series championship in 2010.
He has recorded 171 saves and a 3.21 earned run average in 315 appearances in his seven seasons with the team.