In the event the White Sox would retain A.J. Pierzynski, it's hard to see how the Sox could keep Tyler Flowers in the backup role. He's worthy of part-time duties, not barely-any-time duties, and the Sox have said as much.
They might not be the only team that thinks so, either. According to the New York Post:
The Mets could also go the trade route for a catcher, as the team seeks either a starter or, at minimum, a right-handed bat to pair with Josh Thole. White Sox backup Tyler Flowers is of some interest to the Mets, according to a source. But the White Sox also might not be in a hurry to deal a catcher with last year’s starter, A.J. Pierzynski, on the free-agent market.