Odor hit .144 in 29 games for Texas before being optioned to Triple-A Round Rock on Monday. He struck out 25 times in 90 at-bats.
Texas purchased the contract of infielder Thomas Field, who was listed in the starting lineup hitting eighth and playing second base.
Odor was the youngest player in the majors at 20 years old when he made his major league debut last May. He hit .259 while playing 114 games in 2014.
Field hit .298 with five homers and 13 RBIs in 29 games at Round Rock. The Texas native played in 33 major league games previously, in 2011-12 with Colorado and 2013 with the Los Angeles Angels.