A GRUMMAN GIV HAS AN ENCOUNTER WITH OPPOSITE DIRECTION ACR DURING A CLB TO FL280 N OF SUS; MO.

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I WAS THE CAPT ON GULFSTREAM IV; ON JUN/XA/99; ON A FLT FROM SUS TO MDW. IT WAS A CLR VFR DAY. APPROX XA59 AND WE WERE CLBING N OUT OF SUS ON A VECTOR. I HAD JUST RECEIVED CLRNC TO CLB TO FL280 AND WAS IN THE CLB WHEN I NOTICED AN ACFT ON THE TCASII ABOUT 500 FT ABOVE US. I ASKED THE COPLT HOW FAR AWAY THE ACFT WAS SINCE WE WERE CLBING THROUGH HIS ALT. THE COPLT SAID ABOUT 12 MI. SINCE THE GIV HAS SUCH A GOOD CLB RATE; I FELT THAT WE WOULD BE CLR OF HIM WITH A NORMAL CLB. SHORTLY THEREAFTER; I GOT A VISUAL ON THE AIRLINER AT OUR 3-4 O'CLOCK POS. AT THIS TIME WE WERE ABOVE THE AIRLINER; HOWEVER; THE AIRLINE CALLED CTR AND SAID THAT HE HAD AN ACFT ALERT AND ABOUT THIS TIME WE GOT A YELLOW WARNING. I COULD SEE THAT WE WERE ABOVE THE AIRLINER AND I WAS CONTINUING MY CLB AT WHICH TIME CTR CALLED US AND TOLD US TO DSND. I COULD SEE WE WERE ABOVE THE AIRLINER AND KNEW THAT WE COULD NOT DSND BUT I DID STOP MY CLB AND LEVELED OFF. AT ABOUT THIS TIME WE GOT OUR RA RED ALERT TO CLB. SINCE I COULD SEE THE AIRLINER WAS BELOW US; I MAINTAINED MY ALT UNTIL WE GOT AN ALL CLR FROM THE TCASII AND THEN I DSNDED BACK TO THE ALT THAT CTR HAD RECLRED TO FL230. I FEEL THAT NO SAFETY WAS COMPROMISED; HOWEVER; IF I HAD INCREASED MY CLB WHEN I SAW THE AIRLINER AND THEN WE WOULD HAVE BEEN WELL CLR AND NONE OF THIS WOULD HAVE HAPPENED.