A pair of openings in the rotation were created earlier this week when Drew Hutchison and Kyle Drabek were placed on the 15-day disabled list with right elbow injuries.
A trio of Minor Leaguers are being considered for Tuesday's outing, while Wednesday likely will be handed to either Carlos Villanueva or Aaron Laffey.
"Any time you lose three-fifths of your rotation, that's never a good series -- or homestand, or road trip, or wherever you are," Blue Jays manager John Farrell said of losing three pitchers in a span of four games.
Right-hander Jesse Chavez appears to be the early favorite to make Tuesday's start. He has arguably been Triple-A Las Vegas' most valuable pitcher this season, posting a 7-2 mark and 3.84 ERA in the hitter-friendly Pacific Coast League.
Fellow Minor Leaguers Joel Carreno and Scott Richmond also are in the mix, but both appear to be long shots at this point. An announcement is not expected until the club travels to Milwaukee for the start of a three-game series on Monday.
This won't be a short-term fix, either, as Drabek and Hutchison -- along with right-hander Brandon Morrow (strained left oblique) -- aren't expected back any time soon.
"All three of these are lengthy if you ask me, unfortunately," Farrell said.