A320 ON VISUAL APCH TO RWY 1L AT MCI EXPERIENCED TCAS RA WITH TFC LNDG RWY 1R; NO SEPARATION EVENT RECORDED.

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WE WERE CLRED A VISUAL APCH TO RWY 1L AT MCI. WE APCHED FROM THE W ON APPROX A 45 DEG ANGLE TO THE RWY CTRLINE ABOUT 3 MI OUTSIDE THE FAF. WE WERE SLIGHTLY HIGH AND DSNDING WITH GEAR AND FLAPS 1 DEG; AUTOPLT/AUTOTHRUST OFF. APPROX 3500-4000 FT AGL; L OF CTRLINE; WE RECEIVED A TCAS TFC ALERT TO OUR R -700 FT BELOW. I ASSUMED IT WAS TFC TO RWY 1R AND TURNED FURTHER TO THE L TOWARDS THE FAF. VERY QUICKLY THE TFC ALERT BECAME AN RA 'MONITOR VERT SPD;' IE; DECREASE DSCNT RATE. I BEGAN TO PULL THE NOSE UP TO GET OUT OF THE RED ZONE (ABOUT LEVEL FLT) WHEN THE RA WENT TO 'CLB.' IT RAPIDLY WENT TO 'INCREASE CLB' WITH THE RED ZONE SHOWING APPROX 1500-2000 FPM CLB. THE FO WAS LOOKING FOR THE INTRUDER BUT COULD NOT ACQUIRE HIM. AT THE INCREASE CLB COMMAND; I SELECTED TOGA AND RAISED THE NOSE AGGRESSIVELY TO ATTEMPT TO GET INTO THE GREEN ZONE NOW VERY NEAR THE TOP OF THE SCALE. AT ONE POINT THE RA SYMBOL INDICATED -300 FT AND VERY NEAR OUR ACFT. THE EVENT CLRED RAPIDLY AND WE WERE STILL ABLE TO COMPLETE THE APCH WELL WITHIN STABILIZED PARAMETERS. WE RPTED EVENT TO APCH WHO SAID 'LOOKS LIKE THAT OVERSHOT A LITTLE.' GND CTL TOLD US TO CALL ON PHONE. WE DID; AND HE DIDN'T REALLY SAY MUCH EXCEPT THEY WOULD DO AN IN-HOUSE REVIEW. I REALLY DON'T KNOW HOW CLOSE WE CAME TO OTHER ACFT. HE WAS ACQUIRED BANKING SHARPLY AWAY AND MAY HAVE GONE UNDER OUR ACFT.