American Airlines said Thursday that pilots have come forward and agreed to staff all but "a few hundred" flights scheduled during the December holidays, as it sought to ease concerns of widespread disruptions during one of the busiest travel times.

The union representing American's pilots, however, responded that "thousands of flights" still remained without full pilot crews.

Just ahead of the year-end travel crush, a glitch in American's computer scheduling system — the one on which pilots bid for time off based on seniority — left more than 15,000 flights without pilot crews between Dec. 17 and Dec. 31, the pilots' union said Wednesday.

After an airline update Thursday saying that "only a few hundred flights" still do not have pilots assigned, the Allied Pilots Association said in an emailed statement that it "is able to view in real time December flight-crew assignments" in the scheduling system. "That data does not support management's statement," the union said.  "In fact, 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 pilots group said.

American, which operates a hub and 390 daily flights in Philadelphia, became aware of the malfunction Nov. 24 and has been scrambling to resolve the problem.

On Thursday, American issued an update saying that the "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."

"In addition," airline spokeswoman Victoria Lupica said, "we have more reserve pilots on hand in December than normal months, and they provide us with the ability to fly many of the trips that are currently uncovered. We have not canceled any scheduled flights in December and will continue to work to ensure both our pilots and our customers are cared for."

American, the world's largest airline, said it would pay pilots 150 percent of their normal hourly wage to cover the flights. The union has filed a grievance, saying the company's offer of premium pay was conditional on "certain circumstances" and did not conform with the labor contract.

"We certainly don't want to see a bunch of passengers sitting around terminals, rather than on their merry way," Gregg Overman, spokesman for the Allied Pilots Association, said Wednesday.  "I think it's one of those 'stay tuned' and hopefully we can come up with a solution."

Affected by the computer glitch were flights originating in Philadelphia, Dallas-Fort Worth, Boston, Miami, New York, Salt Lake City, and Charlotte, N.C., according to a company memo to the union reported by Bloomberg News on Wednesday. American will operate 200,000 flights in December.

"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," Lupica said Wednesday. "We expect to be able to avoid any cancellations."