A system scheduling error has left thousands of American Airlines flights in December without scheduled pilots, the carrier's pilots union said as the airline heads into one of the busiest travel periods of the year.

The Allied Pilots Association estimated that more than 15,000 flights from Dec. 17 to 31 were affected by a glitch in the system in which pilots bid for time off based on seniority.

"Basically there's a crisis at American for manning the cockpits," Dennis Tajer, a spokesman for the Allied Pilots Association, said on Wednesday.

The system error was disclosed to pilots on Friday, the union said.

"We have reserve pilots to help cover flying in December, and we are paying pilots who pick up certain open trips 150 percent of their hourly rate - as much as we are allowed to pay them per the contract," American Airlines spokesman Matt Miller said on Wednesday.

In an email to employees, American had offered pilots the extra pay to work certain flights in the holiday period. But the union filed a grievance, saying that some restrictions on overtime pay violated the group's contract.

The union said on Wednesday that management had still not reached out to discuss how best to resolve the shortage.

"I'm watching a 'Grinch that stole Christmas' thing happening. And we don't want to see that happening for our passengers," Tajer said.

American is hoping to avoid having to cancel flights, which, in addition to being a nuisance for travelers, could cost the carrier millions in lost revenue.

In the best-case scenario, labor costs for the quarter will likely climb as a result of pilots' higher pay on those flights, just months after a substantial mid-contract pay increase for pilots and flight attendants spooked investors and temporarily sent the carrier's shares tumbling.

Shares of American were mostly flat in Wednesday afternoon trading.