Paul mentioned what I understand to be the three main candidates. I'm not sure if there is an order of preference, but ...
Olympia Fields wants it because it gets the club in the spotlight, and because Amos Alonzo Stagg, one of the founders (and the original athletic director and football coach at Chicago, the "Monsters of the Midway") was an original backer of the Olympics. He and Baron Pierre deCoubertin corresponded on the prospects of the Olympics being supported by U.S. amateur sports before they resumed in 1896.
Medinah wants it because it won't have the PGA in 2016. Maybe 2018, the 100th PGA, if everything goes right.
Cog Hill wants it because it would be a golf happening in Chicago. You can park the world across the street from Dubsdread.
Additionally, Conway Farms wants it. There's almost no room for the needed infrastructure (TV compound, corporate tents, etc) at Conway, so I can't see it happening there. Olympia, Medinah and Cog have abundant room.
The preliminary plan is to play the two tournaments back-to-back, much like the Open and Women's Open at Pinehurst No. 2 in a few years. So I can't see two courses being picked.
I also can't see an Olympics in Chicago without the requisite palm-greasing. If Chicago wins the big show, my question is, which total will be higher, the number of golf medals the U.S. wins or the number of indictments eventually handed down for Olympics-related construction, bribery, etc. As Mike Royko once wrote, the city motto, "Urbs in Horto," translates to "Where's Mine?"