Look for an unexpected team to come out of nowhere, like the White sox or Mariners.
I think that the Red Sox are going to be the front runners. Without Jay Bay, their offense is definitely going to take a hit. Gonzalez is young as well, which would bode well for the Sox if they wanted to sign him for a long term deal. I wouldn't be a bit surprised if they offered Clay Buckholz, a few other prospects and a boat load of cash.
I don't see the White Sox in the hunt with Konerko at first base. Seattle has a chance if Kotchman doens't produce. the problem is money and minor league prospects in terms of Seattle snagging him. I don't think that they have the resources.
an unexpected team will be the yankees. remember they picked up xavier nady for no reason when they already had a great outfield? yeah someone will get injured for them and then of course the yankees will trade some prospects to the padres and then he is on the yankees. probably not for more than 2 years though cause eventually the money will come up to get the yankees.
He will still be on the Padres in June. If a trade does happen It might happen in July.
He stays put. the Padres can ill afford to lose their "entire" offense.
The Yankees will get him if anyone does, Red Sox maybe in running-the Padres want much for him.