Are you moving to the city of Chicago or the suburbs?

Info about the Chicago Public Schools gifted schools (must live in Chicago city limits):
http://cps.edu/Schools/Elementary_schools/Pages/Regionalgiftedcenter.aspx

CPS will require testing/exams to qualify, much like a private gifted school.

If you are looking in the suburbs, you can view the thread I started asking about suburban public school gifted programs: http://giftedissues.davidsongifted....an_Chicago_Public_School.html#Post233792

As far as private goes, some private gifted schools include Avery Coonley (Downers Grove), Quest Academy (Palatine), Science & Arts Academy (Des Plaines), and IMSA for high school as indigo suggested above.