A scheduling glitch that allowed American Airlines pilots to take vacation at the same time has left thousands of flights during the busy holiday travel period next month without pilots assigned to them.

American Airlines, the largest U.S. airline, is now offering pilots 150 percent of hourly pay to work those dates, an airline spokesman said.

The scheduling problem has left flights from some of American Airlines' biggest hubs without pilots including Miami, Dallas-Fort Worth, Chicago's O'Hare and New York's LaGuardia Airport, without pilots, according to a memo from the airline to pilots, which was seen by CNBC.

The union estimates some 15,000 flights next month don't have pilots assigned to fly them.

Pilots loaded up their schedules with flights in early December but many opted to take days off around the holidays, after the system allowed it.

"The airline is a 24/7 op," said Dennis Tajer, an American Airlines captain and a spokesman for the pilot's union. "The system went from responsibly scheduling everybody to becoming Santa Claus to everyone.

"The computer said, 'Hey ya'll. You want the days off? You got it.'"

American Airlines spokesman Matt Miller said the airline's employees are "working diligently to address the issue and expect to avoid cancellations this holiday season.

"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," he said. "We will work with the APA to take care of our pilots and ensure we get our customers to where they need to go over the holidays."

The Allied Pilots Association, which represents some 15,000 American Airlines pilots, said the airline notified the union about a problem with the pilots' schedule bidding system last Friday.

The union has filed a grievance, saying the airline's solution violates the pilots' contract because of how the airline restricted overtime pay.

Budget airline Ryanair in September canceled some 2,000 flights, stranding thousands of travelers, due to pilot scheduling problems.

It was unclear how much the scheduling problem will cost American Airlines, if anything. But higher labor costs have concerned investors in the past. American Airlines shares tumbled after the airline announced pay raises for pilots and flight attendants earlier this year.

Shares of American Airlines gave up earlier gains after reports of the glitch, ending the day up less than 0.1 percent. Competitor United Airlines rose 3.2 percent, while Delta Air Lines added 3 percent.