Loney, who was competing for the backup first baseman job behind Mike Napoli, batted .174 during spring training.
The 32-year-old signed a minor-league contract with Texas in January, but the team signed Napoli a month later.
Last year, Loney batted .265 in 366 plate appearances for the New York Mets. Loney spent his first seven seasons with the Los Angeles Dodgers, who traded him to the Boston Red Sox in 2012. He spent the next three seasons with the Tampa Bay Rays before joining the Mets.
In 1,443 regular season games, Loney has a .284 batting average with 108 home runs and 669 RBIs.