The team is listening to inquiries about both players, White Sox general manager Rick Hahn told ESPN's Jerry Crasnick. However, that doesn't mean a deal is imminent or even a lock to happen.
"At some point, not necessarily this offseason, we have to make the decision: Are we best served by extending them through what we project to be the bulk of our (competitive) window, or are we better off making a move like some of the others we've made and trying to reinforce the future with prospects?" Hahn said.
Neither Abreu, 30, nor Garcia, 26, are ancient by any means and both are under team control through 2019.
Over four seasons with the White Sox, Abreu has hit .301/.359/.524 with 124 home runs and 410 RBIs. He also hit a career-high 43 doubles in 2017.
Garcia broke out in a big way in 2017. Long seeming to be on the cusp of stardom, the outfielder slashed .330/.380/.506 with 18 home runs in 136 games. Only Houston Astros second baseman Jose Altuve posted a better batting average in the American League (.346).
While both could conceivably be productive parts of the team's rebuild, they would also fetch solid returns on the trade market.