The move Thursday came a day after Eyre left the game while facing the first batter in the 10th inning of the Rangers' 8-7 loss to Toronto in 11 innings.
The right-hander had pitched a scoreless ninth before aggravating the injury that forced him to start the season on the DL.
In three relief appearances, Eyre has allowed three earned runs in 4 1/3 innings. The 30-year-old missed all of last season after undergoing elbow reconstruction surgery in 2007.
Mendoza is 2-0 in two starts for Oklahoma. The right-hander has 13 strikeouts and six walks in 10 1/3 scoreless innings.