The news was first reported Friday by USA Today Sports.
Woodward, 42, will take on his first managerial job after three years with the Dodgers, who won the National League pennant the past two seasons.
A former utility infielder, Woodward played in parts of 12 major league seasons with the Blue Jays, Mets, Braves, Mariners and Red Sox.
After retiring in 2012, he joined the Mariners organization and served as infield coach from 2014 to ’15 before signing on with the Dodgers.
The Rangers are coming off a dismal 67-95 season that saw them finish at the bottom of the American League West.
Woodward will replace Jeff Banister, whom Texas fired in late September as the team stumbled to its first consecutive losing records in 10 years, after winning division titles in each of Banister’s first two seasons.