American Airlines on Thursday sought to reassure nervous passengers that the majority of flights scheduled during the busy December travel season will operate as planned — despite a computer glitch that left more than 15,000 flights without pilots and in danger of being canceled.

“Out of the 200,000 flights American will operate in December, only a few hundred are currently unassigned to pilots,” airline spokesman Matt Miller said. “That number of open flights continues to decrease thanks to our pilots who are stepping up to the plate and picking up trips to ensure customers are taken care of.”

Miller added that the airline generally keeps a higher number of reserve pilots on hand in December, which will enable them to cover flights that are understaffed.

However a spokesman for the Allied Pilots Association, which represents 15,000 American Airlines pilots, disputed the number.

In a statement posted on its website, APA said it had checked the flight crew assignments and found that “thousands of flights are still listed as unassigned.”

“We remain seriously concerned about the potential for significant schedule disruption for our passengers, pilots, and fellow employees during the critical holiday travel season,” the statement says.

According to a company memo obtained by The Washington Post, the bulk of the flights affected are at the airline’s biggest hub in Dallas-Fort Worth. But flights at Washington’s Reagan National, Boston’s Logan International, Miami International, New York’s LaGuardia, Philadelphia International, Salt Lake City International and Charlotte Douglas International airports are also affected.

APA said Tuesday it was informed of the scheduling issues Friday and was told the airline planned to pay the pilots 150 percent of their normal hourly rate if they would agree to cover the unassigned shifts. The union, however, cautioned pilots about taking the deal, saying in a statement that it could not guarantee that the pilots would actually be paid the higher rate.

Miller declined to say how many flights were affected by the scheduling problems, but Capt. Dennis Tajer, an APA spokesman, said more than 15,000 were affected. American operates roughly 6,700 flights a day. Most of the flights affected are for travel between Dec. 17 and Dec. 31.

Passengers concerned about their reservations can call American at 800-433-7300.