To make room on the 40-man roster, the Jays also announced that right-hander Shane Bieber would be placed on the 60-day injured list.
Sosa, 26, is batting .212/.212/.303 with three RBI in 33 plate appearances and 12 games played with Chicago this season.
Last season, in 544 plate appearances over 140 games played with the White Sox, the Venezuela native hit .264/.293/.434 with 22 homers and 75 RBI.
He mostly played second base for the White Sox, but has seen time at both first and third.
Bieber has yet to make his 2026 appearance for Toronto, dealing with inflammation in his throwing elbow.
The 30-year-old made seven regular-season starts for the Blue Jays last season, throwing 40.1 innings while striking out 37 and posting a 3.57 ERA.
Rich, 18, hasn’t made his professional debut yet. He was taken in the 17th round by the Blue Jays in the 2025 First Year Player Draft out of American Heritage High School.