The White Sox and Red Sox completed a blockbuster trade last offseason that sent ace Chris Sale from Chicago to Boston for infielder Yoan Moncada, pitcher Michael Kopech and others.
Chicago has stripped its team down significantly over the past year, and it is firmly entrenched in rebuilding mode.
Trading the 30-year-old Abreu would be a logical move since he has two years of arbitration at hand before becoming a free agent in 2020, per Spotrac.
Abreu is a one-time All-Star who has produced consistently during his four-year MLB career with at least a .290 batting average, 25 home runs and 100 RBI in each season.
Last season, the Cuban star hit .304 with 33 homers and 102 RBI, which represented his best production since being named American League Rookie of the Year in 2014.
Acquiring Abreu would have been a major coup for a Boston team that struggled offensively at times in 2017.
Despite winning the American League East crown, the Red Sox ranked 27th in Major League Baseball with just 168 home runs.
Unless Boston is able to address first base via trade or free agency, it is set to enter 2018 with veteran Hanley Ramirez as the top option at the position.