The 30-year-old was released by the Pittsburgh Pirates on Monday after he was reassigned to the team's alternate training site following spring training.
Goodwin is a lifetime .250/.317/.455 hitter with 42 home runs and 129 RBIs in five big-league seasons. His best year came with the Los Angeles Angels in 2019 when he belted 17 homers and posted a .796 OPS.
Jimenez is expected to miss five-to-six months following surgery on April 1.
The White Sox placed Robert on the 10-day injured list Tuesday after he suffered an injury Sunday against the Cleveland Indians.