Bird has a 6.75 ERA with a save in five relief appearances for the Rangers in his first big league action.
Texas will make a corresponding move before Friday's series opener against Houston to fill Bird's spot on the 25-man roster. The Rangers were off Thursday after sweeping a three-game series against the Los Angeles Angels.
The Rangers got Bird from the Tampa Bay Rays during the offseason as part of a three-team deal that sent infielder Jurickson Profar to Oakland.
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