Thompson signed a minor-league contract with the Cubs on Wednesday, Tommy Birch of the Des Moines Register reports.
The Mets released Thompson on Wednesday, Will Sammon of The Athletic reports.
The Mets signed Thompson to a minor-league contract Wednesday with an invitation to major-league spring training, Andy Martino of SNY.tv reports.
Thompson elected free agency Monday.
The White Sox outrighted Thompson to Triple-A Charlotte on Friday.
Thompson went 2-for-4 with a double and a two-run homer in Wednesday's 5-4 victory over the Mariners.
The White Sox activated Thompson (oblique) from the 60-day injured list Thursday.
The Dodgers traded Thompson (oblique), Nick Nastrini and Jordan Leasure to the White Sox on Friday in exchange for Lance Lynn and Joe Kelly, Jeff Passan of ESPN.com reports.
Thompson (oblique) began a rehab assignment with the Dodgers' Arizona Complex League affiliate on Friday.
The Dodgers transferred Thompson (oblique) from the 15-day injured list to the 60-day IL on Sunday, Bill Plunkett of The Orange County Register reports.