According to Ken Rosenthal, the White Sox could enter the Alex Gordon market.
Sources: #WhiteSox showing interest in Alex Gordon, among others. Gordon appeals for several reasons: Knows division, LHH, defense, makeup.— Ken Rosenthal (@Ken_Rosenthal) December 22, 2015
Heading into 2016, the White Sox have Melky Cabrera, J.B. Shuck, Adam Eaton, Leury Garcia, and Avisail Garcia slated for outfield duty. While that's not a terrible alignment, it could undoubtedly be improved. Cabrera had a down year in 2015, and Garcia (Avisail), has yet to reach the potential that the White Sox (and at one point the Tigers) saw in him.
As a result, Chicago would be wise to add another outfielder if they can stomach the financial cost; and Gordon would make perfect sense.
Over the course of his career, he's been a fantastic defensive left fielder, which the White Sox desperately need, as they had the 29th worst outfield group by UZR, and UZR/150. Gordon on the other hand has posted positive values in both categories since the 2010 season, and can be counted on going forward.
Signing Gordon would also alleviate some of the pressure on Cabrera who was a below league average hitter for most of the season. The ultimate question will be how much money the White Sox have to play with for 2016, as Gordon could command a contract north of $90 million, for four or five years.
If that's too expensive for Chicago, there are still alternative options available. Dexter Fowler, Denard Span, and Gerardo Parra are all on the market, and will likely sign for less than Gordon. The latter two, Span and Parra, don't have a qualifying offer attached, and would allow the White Sox to keep their 2nd round pick, as their first is already protected.
Nothing appears imminent at the moment, but if the White Sox are truly interested in Gordon, it's fair to assume that they'd have an interest in other free agent outfielders.