The nine-year veteran spent last season with the Mariners, posting an 11-16 record and a 5.26 ERA in 32 starts. The Mariners declined their 2014 option on Saunders' contract after the season.
Saunders, an All-Star in 2008 with the Angels, has pitched for three teams -- the D-backs, Orioles and Mariners -- over the past two seasons. He is 89-81 with a 4.30 ERA in his career.
A report on Tuesday said that Saunders worked out for the Rangers last week in Surprise, Ariz., and the lefty told MLB Network Radio that Texas would be a "perfect fit." Saunders adds depth to a starting rotation that may not have Derek Holland and Matt Harrison at the beginning of the year because of injuries.