An air carrier pilot reported during engine start after push back ramp personnel reported a fire from one of the engines. Pilot shut down engine and aircraft was towed back to gate.

Narrative

Pushed back from the gate. Tried to start the engines the temperature this morning was -23 degrees. The fans were barely turning. Couldn't get engine 1 started. Tried to start the other one and rampers on the headphones started screaming fire fire. Shut it down ran the QRC and subsequent checklist asked the rampers what they saw told the FAs (Flight Attendants) what had happened and called Ops and Maintenance Control. Took them 45 to tow us back to the gate. Snow and ice everywhere; plane; ramp; tugs. Finally get a super tug to get us back to the gate. Maintenance wants to start the engines at the gate; told me looks like number two never started. I pull everyone off the plane; we tried to start number one again at the gate; didn't go. Then we tried to start number 2 and then the EGT (Exhaust Gas Temperature) spike and we had an EGT overtemp and exceedance. Maintenance took it from there. We talked with ops and delayed us till XH:50 AM for an aircraft swap. Since I was up at close to XA:00 AM I didn't feel like I would be safe to operate being delayed further. called scheduling to report fatigued.