Texas failed to offer Rosales a contract by the Dec. 2 deadline, allowing the infielder to become a free agent. The 31-year-old hit .262 last year with four homers and 19 RBIs in 181 plate appearances. He had a $750,000 salary and earned a $25,000 performance bonus.
Rosales also has played for Cincinnati and Oakland in seven big league seasons.
To clear a roster spot, the Rangers designated right-hander Matt West for assignment.
Texas also agreed to a minor league contract with 33-year-old right-hander Juan Carlos Oviedo, who was 3-3 with a 3.69 ERA in 32 relief appearances for Tampa Bay last season in his return from Tommy John surgery.
Oviedo had 92 saves for the Miami Marlins from 2009-11, when he was known as Leo Nunez.