The injury comes one day after Borbon was removed in the second inning of a game against Colorado for what Texas manager Ron Washington said was a "teachable moment."
Borbon is hitting .375 this spring with five stolen bases, but also has committed five errors in 19 games.
In addition, Texas Rangers pitcher C.J. Wilson left his latest spring start after two innings because of tightness in his left hamstring.
Wilson left Saturday's game against the Chicago Cubs as a precautionary measure. He then went to the back fields and completed his work in a bullpen session.
Wilson is set to start on opening day next week against Boston. The lefty gave up three runs and four hits against the Cubs before coming out.