Calhoun, 27, had been optioned to Triple-A Round Rock early last month after hitting .136 in 18 games with one home run and two RBIs and then requested a trade. He hit .221 with four homers and 15 RBIs with Round Rock.
Calhoun came to Texas with pitcher A.J. Alexy and infielder Brendon Davis on July 31, 2017, for Darvish, then a four-time All-Star with the Rangers.
Calhoun hit .241 in six seasons with Texas. His best season was 2019, when he hit .269 with 21 homers and 48 RBIs.
Calhoun suffered a broken jaw during spring training 2020 when he was hit by a pitch and played only 29 games that year. His 2021 season ended in June when he was hit by a pitch that fractured his left forearm.
Rangers president Jon Daniels said the decision to designate Calhoun wasn't difficult and was unrelated to Calhoun's trade request.
"At some point the opportunities run out," Daniels said.
Calhoun was designated for assignment to make room on the 40-man roster for outfielder Steele Walker, whose contract was selected from Round Rock. The left-handed-hitting Walker, 25, hit .297 in 20 games this season with Round Rock. He was scheduled to make his major league debut Sunday at home against Seattle.
Texas also optioned infielder Andy Ibanez to Round Rock on Sunday.