It has about 15,000 active pilots and expects to operate more than 200,000 flights in December.
American Airlines has reportedly offered pilots extra pay to work certain flights in the busy holiday travel period.
"Basically there's a crisis at American for manning the cockpits", said Dennis Tajer, a spokesman for the Allied Pilots Association.
American Airlines is the largest USA carrier. He claimed the company does not expect any cancellations, but he would not disclose the number of the flights affected.More news: Lukaku escapes three-match ban over Bong clash
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The issue with scheduling left flights from the airline's hubs, including New York, Miami, Dallas-Fort Worth, Chicago O'Hare, and New York LaGuardia without any pilots assigned to fly, according to memo seen by CNBC. The union has also filed a grievance alleging that the higher wage deal the company is now offering would violate a labor pact. 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.
American is offering pilots 150 percent of their normal hourly wage to pick up some of the flights, the top rate laid out in their contract, Miller said.
The APA also said in its statement that it has not been asked to help American find solutions to the problem. As a result, thousands of flights now do not have pilots assigned to fly them during the upcoming critical holiday period.
As of early on Wednesday, the union said management had still not reached out to discuss how best to resolve the shortage.