Desmond was set to take part in workouts Monday after joining the Rangers at spring training.
General manager Jon Daniels called Desmond a "natural fit" for the Rangers.
The Nationals extended a $15.8 million qualifying offer to Desmond, but he elected to test free agency. The Rangers must surrender a first-round draft pick to the Nationals for signing Desmond.
Former Nationals teammate Ryan Zimmerman reacted to Desmond's leaving the team and taking less money.
"It's a tough situation, obviously, and a situation that's kind of hard to believe he was in," he said, according to MASNSports.com. "But that's the way it is, and I'm just happy he got a job and a chance to go out there and do what he loves doing. I would not be surprised if he goes there and does something special for this one year."
After playing almost his entire career at shortstop, the 30-year-old Desmond is expected to play a utility role with the Rangers. Elvis Andrus is Texas' incumbent at shortstop. The Rangers need help in left field because Josh Hamilton is expected to miss at least the first month of the season because of left knee issues. Desmond has started one of his 927 career games in the outfield.
Desmond has a .264 batting average with 110 homers and 432 RBIs in his career. He has been an All-Star once and won three Silver Slugger awards.
He struggled for much of 2015, batting .233 with 19 homers and 62 RBIs. Desmond also made a National League-high 27 errors.
To make room on their 40-man roster, the Rangers placed right-hander Tanner Scheppers on the 60-day disabled list because of torn cartilage in his left knee.
Information from ESPN's Jim Bowden, Jerry Crasnick and The Associated Press was used in this report.