Crawford is batting .316 with 12 homers and 59 RBIs for the Rays, who began the day three games behind Boston for the AL wild card.
With 54 stolen bases, Crawford is tied for the AL lead with Boston's Jacoby Ellsbury.
Crawford left Monday night's 12-7 win over Toronto after three innings with a sore back and was replaced by Willy Aybar. Crawford was examined by medical staff following the game but said he felt no better Tuesday.
"It's 50-50 at best that he'll be able to play tomorrow," Rays manager Joe Maddon said. "I'm preparing mentally that he's not going to play. We'll see."
Crawford said his injury, which is not considered serious, could be the result of a long stretch of games on artificial turf. Following a nine-game homestand, Monday was Tampa Bay's 10th straight game on turf.
The Rays are off Thursday and open a four-game series in Detroit on Friday.
Gabe Kapler started in left field in Crawford's place.