The team announced Sunday its 25-man Opening Day roster ahead of Monday's road game against the Baltimore Orioles.
Upton, 32, hit .194 with three home runs and seven RBIs during spring training. He has 164 home runs and 586 RBIs in 12 major league seasons with a career batting average of .243.
Upton, who was acquired by the Blue Jays last July from the San Diego Padres, hit a combined .238 with 20 homers, 61 RBIs and 27 steals in 149 games. He made 41 outfield starts for Toronto after the trade and hit just .196 with four home runs and 16 RBIs in 57 games.
The Padres are responsible for all but $1 million of Upton's $16.45 million salary in 2017, according to ESPN.
Osuna's DL stint was made retroactive to March 30 with his injury listed as a cervical spasm. The right-hander's fastball velocity decreased by several mph this spring.
Osuna, 22, had 36 saves for the Jays last season with an ERA of 2.68 in 72 appearances.
Veteran right-handed reliever Jason Grilli is expected to be the closer during Osuna's absence.