A B737 WAS DISPATCHED ON A FLT TO ZZZ WITH NO HOLD AND NO ALTERNATE FUEL ON BOARD. THEY WERE TOLD TO EXPECT HOLD OF 20 MINS SO THEY DECLARED MINIMUM FUEL.

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WHEN WE BEGAN OUR DSCNT INTO ZZZ; WE WERE TOLD TO HOLD FOR APPROX 15-20 MINS AND I TOLD THE CTLR THAT WE DIDN'T HAVE HOLDING FUEL AVAILABLE FOR THAT TIME AND WOULD PROBABLY HAVE TO DIVERT. I CONTACTED DISPATCH AND TOLD THEM THE SITUATION AND THOUGHT THAT ZZZ1 OR ZZZ2 WOULD BE THE BEST CHOICES AND ZZZ1 WAS ALMOST RIGHT UNDER US AND WOULD PREFER THAT AND WAS GIVEN THE APPROPRIATE INFO FOR DIVERT TO ZZZ1 (NOTAMS; WX; ETC) AND THEN FO TOLD ME THAT THE HOLDING WOULD BE ONLY 1 TURN AND THAT WE WOULD BE GIVEN CLRNC BACK TOWARD ZZZ IN ABOUT 2-3 MINS. I TOLD THE CTLR THAT IF WE DON'T GET THE TURN BACK TOWARD THE ARPT WE WOULD HAVE TO DECLARE AN EMER DUE TO LOW FUEL. THIS WAS BECAUSE WE SHOULD HAVE STARTED OUR DIVERT TO ZZZ1 PRIOR TO THE TURN BACK TOWARD ZZZ AND THE DELAY TO WAIT FOR THE TURN TO ZZZ WAS GOING TO PUT US IN A LOW FUEL STATE. I DID NOT 'DECLARE AN EMER' ONLY TOLD THAT I WOULD HAVE TO IF WE DIDN'T GET THE TURN BACK IN THE TIME THEY STATED. CONTACTED DISPATCH ON THE GND AND EXPLAINED THAT WE DIDN'T DECLARE AN EMER ONLY THAT WE WOULD HAVE TO IF WE DIDN'T GET THE TURN INTO THE ARPT AS INDICATED; THE DUTY MGR WAS ON THE PHONE AND HE SAID DON'T HAVE TO SEND ANY RPTS SINCE I DIDN'T ACTUALLY DECLARE THE EMER BUT THE CTLRS MISINTERPED MY RESPONSE TO THE TURN BACK TO THE ARPT AND ROLLED THE CFR ANYWAY. SUPPLEMENTAL INFO FROM ACN 711700: I SHOULD MENTION THAT ATC DID ASK THE CAPT FOR SOULS ON BOARD AND FUEL; WHICH HE STATED AS 45 MINS. WE HAD 5.1 IN THE CHOCKS. APCHING ZZZ ON THE ARR; ATC INSTRUCTED US TO HOLD AT INTXN DUE TO VIP HANDLING. THE FLT PLAN SHOWED 3.3 FOR RESERVES. CAPT ELECTED TO INSERT 5.0 IN THE FMS. WE HAD 3 MINS HOLDING AVAILABLE WITH THAT NUMBER. I TOLD ZZZ APCH THAT WE COULD DO ONLY ONE 360 DEG TURN. APCH ASKED FOR ANOTHER 5 MI S. THE CAPT HAD BEEN COORDINATING WITH DISPATCH; AND SAID TO ATC 'IF WE CAN'T TURN N SOON; I WILL HAVE TO DECLARE AN EMER AND DIVERT TO ZZZ1.' I SUGGESTED HE DECLARE MINIMUM FUEL. SHORTLY AFTER; WE WERE TURNED N AND VECTORED TO FINAL. WE NOTICED TRUCKS AROUND DURING MY LNDG; WHICH WAS UNEVENTFUL. I THINK THERE WAS MISUNDERSTANDING.