The 29-year-old relief pitcher is suffering from thoracic spine inflammation.
Swanson's last outing came on Saturday, where he pitched 1/3 of an inning and allowed one hit in an 8-3 win over the Cleveland Guardians.
Toronto snapped a three-game losing skid with the victory and sits 1.5 games behind the Houston Astros for the third and final American League wild-card spot.
The Blue Jays also recalled right-hander Jay Jackson from Triple-A Buffalo.
Jackson will be active for Sunday's series finale against the Guardians.
This report by The Canadian Press was first published Aug. 27, 2023.
The Canadian Press