Cleveland selected Putnam's contract from Triple-A Columbus before Tuesday night's game in Texas.
Putnam, who has never pitched in the majors, went 6-3 with a 3.65 ERA and nine saves in 44 appearances for Columbus.
Manager Manny Acta says Choo has a chance to play in the series finale Thursday.
When Choo gets back in the lineup, Kosuke Fukudome could play left field for the first time. He caught fly balls in left during batting practice Tuesday.