WASHINGTON — The U.S. military reported its biggest day of evacuation flights out of Afghanistan on Monday, but deadly violence that has blocked many desperate evacuees from entering Kabul’s airport persisted, and the Taliban signaled they might soon seek to shut down the evacuation.

Twenty-eight U.S. military flights ferried about 10,400 people to safety out of Taliban-held Afghanistan over the 24 hours that ended early Monday, a White House official said. The chief Pentagon spokesman, John Kirby, said the faster pace of evacuation was due in part to coordination with Taliban commanders on getting evacuees into the airport.

“Thus far, and going forward, it does require constant coordination and deconfliction with the Taliban,” Kirby said.

With just eight days left before an Aug. 31 withdrawal deadline, the Pentagon is ramping up evacuations from Kabul’s airport by deploying American helicopters and troops into select spots in Kabul to extract stranded U.S. citizens and Afghan allies.

U.S. officials said a military helicopter picked up 16 American citizens Monday and brought them onto the airfield for evacuation. This was at least the second such rescue mission beyond the airport. Kirby said three Army helicopters picked up 169 Americans on Thursday near a hotel just beyond the airport gate and flew them onto the airfield.

Defense officials said that as of Monday, the military has helped to evacuate 37,000 people since Aug. 14, when Kabul fell to the Taliban. The pace of flights has picked up in the last few days, allowing for the 10,400 people to be evacuated in one day. But that number is still just a fraction of the U.S. citizens, foreign nationals and Afghan allies who are seeking to leave the country.

President Joe Biden’s national security adviser, Jake Sullivan, said at the White House that talks with the Taliban are continuing.

“We are in talks with the Taliban on a daily basis through both political and security channels,” he said, adding that ultimately it will be Biden’s decision whether to continue military-led evacuation operations beyond Aug. 31.

After more than a week of evacuations plagued by major obstacles, including Taliban forces and crushing crowds that are making approaching the airport difficult and dangerous, the number of people flown out met — and exceeded — U.S. projections for the first time. The count was more than twice the 3,900 flown out in the previous 24 hours on U.S. military planes.

Army Gen. Stephen Lyons, head of U.S. Transportation Command, which manages the military aircraft that are executing the Kabul airlift, told a Pentagon news conference that more than 200 planes are involved, including aerial refueling planes, and that air crews are working round the clock to accelerate the evacuation.

“They’re tired,” Lyons said of the crews. “They’re probably exhausted in some cases.”

On a more positive note, Lyons said that in addition to the widely reported case of an Afghan woman giving birth aboard a U.S. evacuation aircraft, two other babies have been born in similar circumstances. He did not provide details.

The Pentagon said it has added a fourth U.S. military base, in New Jersey, to three others — in Texas, Virginia and Wisconsin — that are prepared to temporarily house arriving Afghans.

Maj. Gen. Hank Williams, the Joint Staff deputy director for regional operations, told reporters there are about 1,200 Afghans at those military bases. The four bases combined are capable of housing up to 25,000 evacuees, Kirby said.

Afghan evacuees continued to arrive at Dulles International Airport outside of Washington. A bus carried some of the latest arrivals from Dulles airport to another site for what would be one of many processing stops before they reach new homes in the United States.

Exhaustion clouded the faces of many of the adults. How does it feel to be here, a journalist asked one man. “We are safe,” he answered.

Biden said Sunday that he would not rule out extending the evacuation beyond Aug. 31, the date he had set for completing the withdrawal of troops and formally ending the nearly 20-year U.S. military role in Afghanistan. And British Prime Minister Boris Johnson plans to press Biden for an extension to get out the maximum number of foreigners and Afghan allies possible.

Biden is to face the G-7 allies Tuesday in a virtual summit on Afghanistan.

But Taliban spokesman Suhail Shaheen told Sky News that Aug. 31 is a “red line” the U.S. must not cross and that extending the American presence would “provoke a reaction.”

Since the Taliban seized the capital, the U.S. has been carrying out the evacuation in coordination with the Taliban, who have held off on attacking under a 2020 withdrawal deal with the Trump administration.

Meanwhile, a firefight just outside the airport killed one Afghan soldier early Monday, German officials said. It was the latest in days of turmoil outside the airport. Afghans hoping to escape Taliban rule face sporadic gunfire, beatings and crowds that have trampled many.

The New York Times contributed.