Rangers general manager Jon Daniels said Monday that Rodriguez provides a starting option and "he threw well in winter ball and again this spring."
The 36-year-old Rodriguez joined Atlanta on a minor league deal in January before his release Friday. He went 2-0 with a 3.13 ERA in six Grapefruit League starts for the Braves this spring.
If called up for his 11th major league season, Rodriguez would be reunited with former Pirates coach Jeff Banister, now Texas' first-year manager. Rodriguez was released by Pittsburgh on June 1 last season after making six starts.