A good reference for this is hoopshype.com. You can check out any team's salaries in the Salaries section on the top menu bar. Also check out wikipedia for a step by step breakdown of the salary cap.
Basically the Bulls are on hook for $63mill nxt season, which means they already exceed the salary cap, but since the NBA salary cap is a "soft" cap, teams are allowed to exceed the cap to resign their own players (Larry Bird rights) without penalty. There's also the mid-level exception, bi-annual exception, & rookie exception which can all be used to go over the cap. A team is only penalized when it exceeds the Luxury Tax Threshold which was around $72mill this yr.
So basically if BG signs elsewhere, all the Bulls have is the Mid-Level, Bi-annual, & rookie exceptions. But they will already be over the cap anyway at $63 & the LT Threshold is going to be less next season too. So there's really not much the Bulls can do in FA season.