Cueto said Sunday that he was not worried about the stiffness in his elbow after being scratched from his scheduled outing against the Cleveland Indians.
"He hasn't had total relief, and I'd have thought by now he would," Price said.
Cueto, who won 20 games last season, is in the final year on his contract. The Reds had lost eight games in a row heading into Monday's contest and likely would consider trading the right-hander at midseason if they're still far out of the race.
The Associated Press contributed to this report.