The move is retroactive to Sunday.
Choo, who didn't make the trip to Seattle for the team's series with the Mariners, will undergo surgery in the next two weeks. The operation is set for Sept. 15 but could be moved up to Friday if Choo's illness subsides. He missed Sunday's game against Kansas City with the flu.
"By the end of October, he would be 100 percent and have a completely normal ... offseason," general manager Jon Daniels said, explaining why the last-place Rangers didn't want to wait until after the season. "We want everything, from a health standpoint, we want 100 percent for 2015."
Choo, 32, struggled in his first season with Texas after signing a $130 million, seven-year contract when he was a free agent last winter. He hit .242 with 13 homers and 40 RBIs in 123 games.
He also played through an ankle injury as well as the elbow issue, which hurt him on long throws from the outfield, Daniels said. Choo appeared in only 75 games in the outfield and made 44 starts at designated hitter.
The arthroscopic procedure will be performed by Dr. Keith Meister, the Rangers' team physician.
Outfielder Michael Choice was recalled from Triple-A Round Rock to replace Choo.
Choice is batting .267 with seven home runs and 31 RBIs in 43 games for Round Rock. He started the season with the Rangers and hit .177 with eight homers and 28 RBIs in 69 games before getting demoted.
Information from The Associated Press was used in this report.