Sure you can. Mount Sinai in Chicago offers all of the cores. Electives might be harder to come by but Sinai offers three electives, Cook County I believe offers four. But why limit yourself to only one city? There's tons of great hospital systems out there and the more you spread yourself out the more options you will have in getting interviews when it comes time for residency applications.
However, keep in mind that cores are assigned to you by your CC and you have very little say in the matter, unless you are willing to wait for an extended period of time to be where you want. Considering the ever-increasing US medical school matriculants and graduates with the stagnant number of residency positions, it behooves you to graduate earlier for an earlier match. Moving around for your cores and electives, with the prime objective being a timely graduation, is something your mind should get used to.