As the scuffling Chicago White Sox reportedly inch closer to becoming sellers, rumor has it one of their big-name pickups from a year ago could be made available.

The White Sox are pushing to move struggling closer David Robertson ahead of Monday's trade deadline, sources tell ESPN Insider's Buster Olney, and the Boston Red Sox are among several teams believed to have interest.

Olney reports the Red Sox have discussed dealing for Robertson internally, though it's unclear whether that dialogue has advanced into substantial talks with the White Sox. Boston's currently without injured relievers Koji Uehara and Craig Kimbrel, who is expected to return to the club Monday following a speedy recovery from knee surgery.

Robertson has struggled in his second season on the South Side, authoring a 4.35 ERA with his highest walk rate (11.6 percent) in five years. An All-Star in 2011 with the Yankees, the 31-year-old Robertson is under contract for two more seasons at a total of $25 million.

It was reported earlier Thursday that the White Sox were still receiving offers for ace Chris Sale, who's being pursued by the Texas Rangers as they look to acquire a big-name pitcher.

The White Sox sat eight games behind division-leading Cleveland and six back of the second wild-card spot entering Thursday's home-and-home finale against the Cubs.