ATC GAVE AN IFR MLG CLB INSTRUCTIONS TO AVOID A VFR SMT.

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DURING CLBOUT FROM JFK ENRTE TO CLE; FLT EXPERIENCED THE NEAR MISS AT 7400 FT MSL; 15 NM; 235 DEGS FROM JFK VOR AT XA35 LCL. TALKING TO NY DEP ON 135.9 MHZ; WE WERE GIVEN A CLB TO J110 DIRECT RBV VOR ON COURSE. THROUGH 6800 FT GIVEN TURN TO 290 DEGS (30 DEGS R OF PREVIOUS COURSE). ON THE NEW HDG; WE WERE ISSUED A TA 'VFR TFC 11 O'CLOCK 7500 FT EBOUND LESS THAN A MI.' AT THAT POINT (TFC NOT IN SIGHT) A NEW CTLR (SUPVR?) ISSUED 'EXPEDITE CLB THROUGH 8000 FT FOR 11000 FT; CLRED DIRECT RBV.' IN THE CLBING L TURN WE SPOTTED THE LIGHT TWIN AT 12 O'CLOCK; APPROX 400 FT BELOW OUR NOSE. WHEN QUERIED ABOUT THE CLOSE CALL THE CTLR THOUGHT A CLB THROUGH THE ALT WOULD RESOLVE THE TFC CONFLICT BETTER THAN A LEVEL OFF BELOW THE ALT (WE DISAGREE!). OUR HDG WAS FINE TO AVOID TFC; THE TURN BACK DIRECT TO RBV PUT US ON A CONVERGING COURSE. IF THE CTLR HAD LEFT US ON THE ORIGINAL HDG WE WOULD HAVE HAD NO CONFLICT. OUR ACFT; UNFORTUNATELY WAS NOT EQUIPPED WITH TCASII. IF WE HAD IT; WE ARE CERTAIN THIS WOULD HAVE BEEN AN RA WITH EVASIVE COMMANDS. THE VFR TFC WAS APPARENTLY NOT TALKING TO APCH WHILE TRANSITING OVER THE CLASS 'B' AIRSPACE. VFR TFC IN; AROUND; OR ABOVE CTLED AIRSPACE (ESPECIALLY IN THE NE CORRIDOR) SHOULD COMMUNICATE WITH ATC REGARDLESS OF THE RTE OR CONDITION OF THE FLT. THIS MAY HAVE HELPED THE SEE AND AVOID PARTNERSHIP WE ALL MUST SHARE.