The Rangers placed Winn on the 15-day injured list Sunday due to right arm fatigue.
The arm fatigue may have played a part in Winn's poor showing in a 5-2 loss to the Angels on Saturday, when he retired one of the three batters he faced and was charged with two earned runs on two hits. Texas recalled right-hander Gavin Collyer from Triple-A Round Rock to replace Winn in the bullpen.