American Airlines Booking Phone Number Denver Airport - If you are planning to travel with American Airlines from Denver International Airport (DEN), the booking phone number is the easiest way to handle all your travel needs. With one call, you can book new tickets, confirm existing reservations, change your travel dates, cancel flights or request a refund. Passengers departing from Denver Airport can also get updates on flight schedules, seat upgrades, check-in options and premium class bookings.