Jennings appeared in 10 games and posted a 1.74 ERA, with eight strikeouts and four walks in 10 1/3 innings.
"It's one of those tough decisions that you've got to make," manager Mike Redmond said. "He's going to be fine. He will go down there and pitch, and we know he has value to us. It's a tough decision."
Turner went on the DL on April 9, retroactive to April 4, due to a right shoulder sprain.
Joe Frisaro is a reporter for MLB.com. He writes a blog, called The Fish Pond. Follow him on Twitter