The 36-year-old Kelly played first base Wednesday against the New York Mets, his first major league start since April 12, 2015.
Bour is batting .268 with 15 homers, 46 RBI and a .347 on-base percentage. He rolled his right ankle while racing to first base on a grounder Saturday against Atlanta, and he was sent back to Miami on Tuesday for an MRI.
Kelly had Tommy John surgery last July and was batting .223 in 48 games at New Orleans this year. He is married to the sister of Mets second baseman Neil Walker.
To make room for Kelly on the 40-man roster, the Marlins designated left-hander Eric Jokisch for assignment.