I don't like making predictions about trades, because I always feel silly since I always under or over estimate a player's value to each team involved or miss some crucial factor. But just for fun, and because it is so unlikely and off the radar (so maybe it will be another Friedman stealth attack), here is one.
Why the Rays would do it: Although 35, Konerko has been durable and his last few years are right in line with his career numbers. He walks, does not strike out a lot and has power. His splits do not show a problem with righties or outside his home park. Last year his OPS was well over .800 in both cases, and for his career, he has hit very well at Tropicana. His $12 million 2012 salary would be eased by the money, and the $13.5 million commitment in 2013 is actually $6.5 million with the rest spread out over 7 years at $1 million per year.
Although Flowers has been disappointing, he showed some improvement last season (and catchers often develop late) and he has some pop and walks a lot. I read at Sickels that his defense has improved. He has a chance to become an adequate major league catcher or at least a backup.
Why the White Sox would do it: Stuck with Dunn, he could move to 1B where his offense might rebound. They have lost Buehrle and probably Jackson, and Peavy is a question, so there is room in the rotation. Their farm system is weak, and the recent trade of Santos for Molina indicates that Williams wants to improve there as are rumors that he is shopping Danks and Floyd. They would also save a bit of money on Konerko's salary. If, as is reasonable, they do not expect to contend in 2012, it helps the rebuilding.
Why it won't happen: Chicago probably values Konerko much higher, and it is still quite a bit of money for the Rays to commit. Giving up 3 young players for an old first baseman and a probable backup catcher is not typical of Friedman.