Keep it unloaded and enclosed in a case and you will be in compliance with Illinois State Law (and a lock on the case will avoid all hassle at all because you can refuse police permission to open the locked case):
(430 ILCS 65/2) (from Ch. 38, par. 83‑2)
Sec. 2. Firearm Owner's Identification Card required; exceptions.
(a) (1) No person may acquire or possess any firearm, stun gun, or taser within this State without having in his or her possession a Firearm Owner's Identification Card previously issued in his or her name by the Department of State Police under the provisions of this Act.
(b) The provisions of this Section regarding the possession of firearms, firearm ammunition, stun guns, and tasers do not apply to:
(9) Nonresidents whose firearms are unloaded and enclosed in a case; (or)
(10) Nonresidents who are currently licensed or registered to possess a firearm in their resident state;
(720 ILCS 5/24‑1) (from Ch. 38, par. 24‑1)
(Text of Section from P.A. 96‑41)
Sec. 24‑1. Unlawful Use of Weapons.
(a) A person commits the offense of unlawful use of weapons when he knowingly:
(4) Carries or possesses in any vehicle or concealed on or about his person except when on his land or in his own abode or fixed place of business any pistol, revolver, stun gun or taser or other firearm, except that this subsection (a) (4) does not apply to or affect transportation of weapons that meet one of the following conditions:
(i) are broken down in a non‑functioning state;
(ii) are not immediately accessible