The US says it will hold the Taliban to a promise to let people leave Afghanistan after 31 August.
US Secretary of State Antony Blinken has said the Taliban have promised to allow foreigners and Afghans to leave Afghanistan beyond 31 August, when a US-controlled airlift will end.
He said Washington and its allies had “a responsibility to hold [the Taliban] to that commitment”.
Mr Blinken said the US was still on track to complete operations at Kabul airport by the end of the month.
There are still thousands inside and outside the airport wanting to fly out.
More than 82,000 have been airlifted from Kabul, which fell to the Taliban 10 days ago. The militants have opposed extending the deadline of the US-led evacuation.
Amid scenes of chaos, flights have been stepped up in recent days. About 19,000 people have been evacuated in the past 24 hours, Mr Blinken said.
“Only the United States could organise and execute a mission of this scale and this complexity,” he told reporters in Washington.
“The Taliban have made public and private commitments to provide and permit safe passage for Americans, for third-country nationals and Afghans at risk past August 31st,” he said.
He added that the US would help those who wanted to leave Afghanistan “not just during the duration of our evacuation and relocation mission, but every day thereafter”.
Up to 1,500 US nationals could still be in the country and the administration was making exhaustive efforts to trace them, he said.
The CIA and the US military have been carrying out clandestine operations to extract Americans using helicopters and ground troops, US and other officials have told the Wall Street Journal.
The perilous operations have been both inside and outside of Kabul, the paper says.
Other countries are also rushing to evacuate their nationals, as well as Afghan staff and visa holders, ahead of the deadline.
The UK government said its operation was moving at “significant pace”, with some 1,200 people flown out on Wednesday.
Kabul airport is currently being defended by 5,800 US and 1,000 British troops.