Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Time Pressure; Workload
Aircraft Equipment Problem Less Severe; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Flight Deck / Cabin / Aircraft Event Smoke / Fire / Fumes / Odor
Air Traffic Control Provided Assistance; Flight Crew Landed in Emergency Condition; Flight Crew Overcame Equipment Problem; Flight Crew Requested ATC Assistance / Clarification; General Maintenance Action
Aircraft
About one hour from landing; we started to smell a faint burning scent from the packs. It was very faint; but noticeable. It smelled like glycol or plastic burning. It remained constant until we initiated a descent from FL390-FL370. The smell became more pronounced. For lack of another checklist or procedure; we ran the smoke fire and fumes QRH. The smell was only in the cockpit and not the cabin. Since we could not determine the source. We deemed it appropriate to take the safest course of action to expedite getting the aircraft on the ground with no undue delay. We [advised ATC] and asked for CFR (Crash Fire Rescue) to meet us to check for anything out of the ordinary upon landing. We initiated a high speed descent and approach; and performed a safe landing. CFR met the aircraft and checked it over and we taxied to the gate.
B737-800 First Officer reported a flight deck fumes event during cruise.  The crew ran the appropriate QRH checklist and continued to destination.