A320
Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument
Communication Breakdown; Physiological - Other
Aircraft Equipment Problem Less Severe; Deviation / Discrepancy - Procedural Clearance; Flight Deck / Cabin / Aircraft Event Illness / Injury; Flight Deck / Cabin / Aircraft Event Smoke / Fire / Fumes / Odor
Air Traffic Control Provided Assistance; Flight Crew Requested ATC Assistance / Clarification; Flight Crew Returned To Gate; General Flight Cancelled / Delayed; General Maintenance Action
Aircraft
Operating Aircraft X; ZZZ-ZZZ1 we pushed back from Gate X in ZZZ and were pulled to the top of the alleyway near spot 1.   After starting engine 1 and disconnecting the push back tug; I smelled a slight fume smell.  Wasn't immediately sure if it was a concern or not so I notified the Flight Attendants and asked if they could smell anything unusual in the cabin.  They did not.  Then after starting engine 2 as we taxied up to spot 1; the fume smell got stronger in the flight deck.  The First Officer now stated that he could smell strong odors.  I checked with the Flight Attendants again; and they now said they could start to smell something.  We notified ground control from spot 1 that we had to return to the gate and we shut down engine 2.  Ground control had us taxi from spot 1; Taxiway 1; Taxiway 2; and hold short of Taxiway 3 awaiting a gate assignment.  We notified Ramp Control that we needed to quickly return to the gate because of fume smells.  We had donned O2 masks and tried a few times to get a gate assignment but Ramp Control kept asking multiple times what the issue was after we had already told them.  I then [requested priority handling] with Ground Control; and they replied for us to contact Ramp Control.  I also [requested priority handling] with Ramp Control and started taxiing from Taxiway 4 to Taxiway 5 spot 2 in anticipation getting into Gate 2.  Ground Control then asked if we needed [priority] equipment.  We acknowledged yes send the personnel to check the outside of the aircraft.  We parked at Gate 2; shutdown; and deplaned the passengers as soon as possible.  The Flight attendants had left the aircraft with the passengers by the time I had given my report to Dispatch; Maintenance; and the [airport authorities].The First Officer and I both had dull headaches; burning sore throats.  I notified safety and they said to call the Duty Pilot on call and they would facilitate our being examined by a medical professional.  After talking to the Duty Pilot/Chief Pilot; I was told to call a Paramedic if I was concerned about my health.  Both of us stopped by the Airport station and asked if they could have the Paramedics take our vitals.  After this was done; I drove the First Officer to ER where they examined us both. [The cause was] Maintenance issues with older aircraft.
After pushing back from the gate; we started both engines.  There was a faint smell of exhaust/fumes that went away and returned back even stronger.  The Flight Attendant said that they could smell it in the cabin; however; it was not that strong.  As we were holding on the taxiway awaiting to get back into the ramp; the smell would dissipate and return again.  The smell was very strong.  I started coughing and having a dry throat sensation.  We donned our O2 masks and [advised ATC]. The smell occurred after starting an engine.
Air Carrier flight crew reported a fume event during departure pushback. Priority handling was utilized to return to gate.