Garcia, 26, is coming off a breakout season in which he slashed .330/.380/.506 with 18 home runs and 80 RBIs in 136 games while earning the first All-Star selection of his six-year career.
The outfielder represents the type of right-handed bat the A's are seeking as they plan to move Khris Davis into more of a designated hitter role, per Shea.
Garcia, who is under team control through 2019, is due for a raise this winter after earning $3 million last season. White Sox GM Rick Hahn is willing to move Garcia, according to Chris Kuc of the Chicago Tribune.
"If there's an opportunity to put us in a better position long term, we'll certainly fully explore it," Hahn said, per Kuc. "Everything is in play."
Oakland finished last in the AL West this past season with a 75-87 record.
-- Nick Ostiller