AN EMS ATX ACFT ENTERS THE CLASS B WHILE FLYING IN MARGINAL VFR WX ON A NO FLT PLAN TYPE OP.

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I WAS CALLED OUT ON AN EMER MEDICAL TRANSPORT FROM XXX WITH A RETURN TO A LCL HOSPITAL IN YYY. WE ELECTED TO REMAIN CLR OF THE CLASS B AIRSPACE; SO STARTED EACH CLB AS WE CLRED THE NEXT CLASS B RING. WE OBSERVED THE ARRIVING TFC ON RWY 13; SO CHANGED OUR COURSE TO THE N TO AVOID THE INBOUND TFC. WE DID NOT OBSERVE ANY TFC THAT WAS CLOSE. UPON OUR ARR BACK TO XXX; WE WERE INSTRUCTED TO CONTACT APCH CTL. WHEN I TALKED TO THEM; I WAS TOLD THERE WAS A POSSIBLE CLASS B INCURSION WHICH CAUSED A TCASII ALERT. THE ACFT WITH THE TCASII ALERT THEN CLBED INTO ANOTHER ACFT. DUE TO THE LOW SURFACE VISIBILITY; IT IS POSSIBLE THAT WE BEGAN OUR LAST CLB EARLY. WE THOUGHT WE WERE BEYOND THE 4000 FT RING. WE WERE MONITORING APCH CTL; ON THE VFR FREQ; AS WE USUALLY DO; AND WERE SQUAWKING OUR DISCREET XPONDER CODE. WE WERE NOT UP ON THE INBOUND TFC APCH FREQ; SO HEARD NO TFC CALLS TO US. IN THE FUTURE; WE WILL ALWAYS TRY TO TALK TO APCH CTL. THE IFR ARRIVING CTLR SHOULD ALSO COORDINATE WITH THE VFR CTLR IF THERE IS A CONFLICT. THIS MAY HAVE PREVENTED US FROM CAUSING A TCASII ALERT.