Thousands of American Airlines flights "during the upcoming critical holiday period" don’t currently have pilots assigned to them because of a scheduling glitch, according to the airline’s pilot union.

“On Friday, management disclosed a failure within the pilot schedule bidding system,” read a statement posted to the Allied Pilots Association’s (APA) website on Tuesday. “As a result, thousands of flights currently do not have pilots assigned to fly them during the upcoming critical holiday period.”

American Airlines said in a statement Wednesday that they were "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," the statement 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 airline did not say how many flights were affected or the exact time period of affected flights.

APA represents the 15,000 pilots with American Airlines.

The union statement said in its statement that on Tuesday "management issued an update detailing the 'significant holes' in the operation and unilaterally invoked a solution for crewing affected flights."

The group said it had filed a grievance in response.

The scheduling problem is affecting flights in some of the nation's biggest airports, including New York, Miami, Dallas-Fort Worth and Chicago, according to a memo from the airline to pilots that was seen by CNBC.

While pilots had loaded up their schedules with flights in the beginning of December, many were allowed to take off time around the holidays, CNBC reported.

"The airline is a 24/7 op," Dennis Tajer, an American Airlines captain and union spokesman, told CNBC. "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.'"