CE750 FLT CREW ABORTS TKOF AFTER SIGHTING B737 ABOUT TO CROSS THE RWY AT MDW.

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CLRED FOR TKOF RWY 31C AT MDW. AS WE BEGAN TO ROLL; NOTICED B737 TAXIING AT VERY HIGH SPD FROM L TO R. IT WAS OBVIOUS HE WAS GOING TO CROSS OUR RWY APPROX 150 YARDS FROM DEP END. PF ABORTED TKOF AT THE SAME TIME AS TWR CTLR SAID 'STOP; STOP; STOP!' B737 CONTINUED ACROSS RWY WITHOUT SLOWING DOWN. WE BOTH SAW THIS COMING AND GOT THE AIRPLANE STOPPED WELL SHORT OF THE INTXN. CALLED TWR ON PHONE AFTER TRIP. HE CLAIMED TWR CANCELED OUR TKOF CLRNC IMMEDIATELY AFTER GIVING IT. I AM SPECULATING THAT I WAS ANSWERING TKOF CLRNC ON THE RADIO AT THE SAME TIME HE WAS CANCELING; SO WE BLOCKED EACH OTHER. B737 WAS PROBABLY ON GND CTL AND HEARD NOTHING. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR STATED THAT HE WAS CLRED INTO POS AND HOLD AND SUBSEQUENTLY CLRED FOR TKOF. RPTR STATED THAT ACFT Y WAS XING IN THE AREA OF TXWY A AND TXWY B AND THAT HIS ACFT BARELY GOT ROLLING AND STOPPED PRIOR TO THE TXWY K INTXN. RPTR STATED THAT IN THE TELEPHONE CONVERSATION WITH THE TWR CTLR; THE CTLR DID NOT MENTION HAVING CHKED THE TWR TAPES. RPTR STATED THAT ALL OF HIS ACFT'S LIGHTS WERE ON AND FUNCTIONING PROPERLY AND THAT ACFT Y HAD ALL OF ITS LIGHTS ON EXCEPT FOR LNDG LIGHTS.