Upton hit just .194 with three home runs and seven RBIs this spring. He homered on Saturday in the Blue Jays' 6-4 exhibition victory over the Pittsburgh Pirates at Olympic Stadium in Montreal.
The disabled list move for Osuna was made retroactive to April 1 with the reliever's injury listed as a cervical spasm. Osuna's fastball velocity has notably decreased this spring, with an average speed of 92 to 93 mph. Last season, the right-hander's fastball consistently was timed at 96 mph.
Osuna told reporters last week that he was "OK" and that he wasn't worried about the dip in his velocity. He pitched a scoreless inning with a strikeout in the Blue Jays' 1-1 tie with the Pirates in Montreal on Friday night. The 22-year-old Osuna saved 36 games last season, posting a 2.68 ERA in 72 appearances.
Veteran reliever Jason Grilli is expected to fill the closer's role in Osuna's absence.
The Blue Jays acquired Upton from the San Diego Padres last July. He made 41 outfield starts for Toronto after the trade and hit .196 with four home runs and 16 RBIs in 57 games. He hit .256 with 16 home runs and 45 RBIs in 92 games for San Diego before the trade.
The Padres are responsible for all but $1 million of his $16.45 million salary in 2017.
Upton, 32, has 164 home runs in 12 major league seasons with a career batting average of .243.
The Associated Press contributed to this report.