The White House said Tuesday that the Taliban had promised that civilians could travel safely to the Kabul airport as the US military stepped up its airlift for Americans and Afghans fleeing Islamist group.
"The Taliban have informed us they are prepared to provide the safe passage of civilians to the airport, and we intend to hold them to that commitment," National Security Advisor Jake Sullivan told reporters.
Sullivan said US military transports had carried US citizens, other foreign nationals and Afghans permitted to relocate to the United States out of the country as the Taliban consolidated their control over their country and began planning a new government.
The White House said there were still around 11,000 "self-identified" Americans still in the country.
Despite some reports that people are being harassed and even beaten as they try to leave, Sullivan said "large numbers" of people had been able to get to the Hamid Karzai International Airport.
He said the US side was taking the issue of harassment "in a channel with the Taliban."
At the Pentagon Major General Hank Taylor said that US military officials at the airport had also been in communication with Taliban commanders about ensuring the removal of thousands of US citizens and Afghans would continue safely over the coming days.
"We have had no hostile interactions, no attack and no threat by the Taliban," said Taylor, a senior official with the Pentagon's Joint Chiefs of Staff.
Taylor said that since the airport was reopened early Tuesday following an hours-long closure for security problems, the US military has evacuated close to 800 people, among them 165 Americans, on seven flights.
The US Department of Defense has poured troops into the airport since Saturday to protect the exodus as the Taliban terrorists entered Kabul after a lightning siege across the country and seized power.
Taylor said the number of US troops would rise from 2,500 on Monday to around 4,000 by late Tuesday.
He said the US aimed to increase its airlift to one aircraft an hour so that between 5,000 and 9,000 passengers could be carried out per day.
"We are confident we have taken the right steps to resume safe and orderly operations at the airport," he said.
Some other countries, including Germany and France, have also been able to land aircraft to pick up their nationals and Afghans qualified to travel to those countries.
Taylor spoke a day after security broke down at the airport, with videos showing hundreds of Afghans on the runway trying to impede a giant C-17 transport and clinging to it.
Videos appeared to show two people falling to their deaths from the aircraft after it took off.
Another, according to the Washington Post, was later found dead in a wheel well.
Pentagon spokesman John Kirby said those incidents were being looked into by the US Air Force.
The United States has pledged to accept tens of thousands of people such as translators who worked for US forces and their families, as they fear retribution from the Taliban.
More than a thousand have been evacuated since the operations began three days ago.