CL65 FLT CREW DURING PUSHBACK ARE NOTIFIED THAT TIRE APPEARS TO HAVE LOW PRESSURE BY RAMP PERSONNEL. FLT CREW ELECTS TO CONTINUE FLT; NOSE TIRE BLOWS ON TKOF; DAMAGES ACFT ON LNDG.

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AFTER PUSHBACK DURING DISCONNECT; RAMP CREW NOTIFIED CREW THAT L TIRE LOOKED LOW. PIC QUERIED FO WHO STATED EVERYTHING LOOKED GOOD ON WALKAROUND. CREW THANKED RAMP CREW AND ELECTED TO CONTINUE. AFTER TKOF; HEARD AN ABNORMAL NOISE IN WHEEL WELL. AS WE SWITCHED TO SOCAL APCH; THEY PASSED ON PIREP OF SOMETHING TRAILING THE ACFT. RETURNED FOR A LOW PASS GEAR INSPECTION WITH TWR. TWR RPTED GEAR DOWN; LOCKED AND STRAIGHT; WITH SOMETHING LIKE A RIBBON TRAILING THE NOSE GEAR. CREW RETURNED FOR NORMAL ILS/VISUAL APCH TO RWY 26L AND LNDG. ON ROLLOUT; NOTICED NOISE FROM NOSEWHEEL; SOUNDED LIKE A FLAT TIRE. ROLLED OFF AT TXWY F PAST HOLD SHORT LINE. PARKED AND NORMAL SHUTDOWN. POSTFLT INSPECTION REVEALED L NOSEWHEEL WAS CARCASS ONLY; TREAD WAS COMPLETELY REMOVED. ACFT SUSTAINED MINOR DAMAGE FROM TIRE FOD. DURING POSTFLT DEBRIEFING; CAPT AND FO HAD DIFFERENT IMPRESSIONS OF WHICH TIRE WAS RPTED LOW. ALTHOUGH THE LOW TIRE RPT MAY HAVE HAD NOTHING TO DO WITH THE BLOWN TIRE ON TKOF; THE MOST APPROPRIATE COURSE OF ACTION IN THE FUTURE WOULD BE TO RETURN TO THE GATE FOR A DIRECT INSPECTION BY THE PIC. AT A MINIMUM; THE PIC SHOULD HAVE REQUESTED MORE INFO ABOUT THE APPEARANCE AND DISPOSITION OF THE TIRE FROM THE RAMPERS; TO MAKE A MORE INFORMED DECISION. ALTHOUGH THE CREW WASN'T SUFFERING FROM GET THEREITIS; THE LOGISTICS OF RETURNING TO THE GATE PROBABLY UNCONSCIOUSLY INFLUENCED THE CREW'S DESIRE TO CONTINUE. PAPERWORK FOR A RETURN TO GATE WOULD HAVE BEEN MUCH LESS THAN PAPERWORK POST-INCIDENT.