"He's working through it. If he's not 100 percent, we're not going to pitch him," said Montoyo, who added that he wasn't sure if Giles would be able to pitch Tuesday.
Giles has been dealing with inflammation in his right elbow since June, when he spent 10 days on the injured list.
The right-hander, who earned the nickname "100 Miles Giles" because of the speed of some of his pitches, has impressive numbers this season when he has been able to pitch: 14 saves in 15 opportunities, a 1.54 ERA and 58 strikeouts in 35 innings.
The timing of his latest elbow woes could cost Toronto a chance to pick up a prospect or two before Wednesday's trade deadline.