I've looked at the White Sox prospects and current team and I don't see a match. The Padres have a deep farm system and really don't have a need to reload it. Any of the prospects the White Sox could give up would be in competition with other prospects and not have a likely path to provide value to the 2013 or 2014 teams.
If you look at the Latos deal the team made in the offseason you can kind of see the template. Now maybe Headley doesn't carry the value of Latos, but you still see the objective. Major league ready and near major league ready players that are starters or key parts of the bullpen. Byrnes has hinted at this both in a brief conversation I had with him in the offseason as well as his more recent comments about the trade deadline.
As of right now Chase Headley isn't a free agent until 2015. There's nothing that forces the Padres hand into trading him for less than an ideal return (as opposed to Adrian Gonzalez where the team basically had to take the best offer available or risk losing him to free agency).
You can also look at the value Chase brings to a team. Fangraphs has his WAR at 2.7 for the year. A high pace, but one he matched in 2010. Right now, the White Sox best player is Paul Konerko and his 2012 WAR is equal to Headley. However, Headley is cheaper and younger than Konerko. How much would the White Sox charge for a cheaper, younger Konerko? That may be extreme, but you get my point.