A Nato official said at least 20 people had died in the past seven days in and around the airport. Some were shot and others died in stampedes, witnesses have said.

“Let me be clear, the evacuation of thousands from Kabul is going to be hard and painful” and would have been “no matter when it began”, Biden said in a briefing at the White House on Sunday.

“We have a long way to go and a lot could still go wrong.”

Biden said he had directed the State Department to contact Americans stranded in Afghanistan by phone, email and other means, and the United States had a plan to move them to the airport.

“We’re executing a plan to move groups of these Americans to safety and to safely and effectively move them to the airport compound. For security reasons, I’m not going to go into detail … but I will say again today what I’ve said before: any American who wants to get home will get home.”

Afghan allies of the West and vulnerable Afghans such as women activists and journalists would be helped too, he said.

On Sunday, there were no major injuries as gunmen beat back the crowds, according to witnesses. The Taliban had been “cooperative” about extending the airport perimeter, Biden said.

Asked by a reporter whether the United States would extend an August 31 deadline for evacuations, Biden replied: “Our hope is we will not have to extend but there are going to be discussions I suspect on how far along we are in the process”.

The United States on Sunday sought the help of six commercial airlines to transport people after their evacuation from Afghanistan. Biden said people were being assisted by more than two dozen countries in four continents.

Britain plans to push world leaders to consider new sanctions on the Taliban when the G7 meets, sources said. Biden said he would support that effort, depending on the conduct of the Taliban.

Panicked Afghans have tried to board flights out of Kabul since last weekend, fearing reprisals and a return to a harsh version of Islamic law the Sunni Muslim group exercised while in power two decades ago.

Leaders of the Taliban, who have sought to show a more moderate face since capturing Kabul, have begun talks on forming a government.

The United States and other countries including Britain have brought in several thousand troops to help evacuate foreign citizens and at-risk Afghans from Kabul, but have been careful to avoid clashes with the Taliban.

A Taliban official said: “We are seeking complete clarity on foreign forces’ exit plan”.

Taliban political office spokesman Mohammed Naeem told Saudi-owned Al Hadath TV that talks were ongoing with the United States and other countries. He said al-Qaeda was not present in Afghanistan and the Taliban has no relationship with it.

Afghans who fled in the past week have spoken about their despair at leaving loved ones behind. “It was very difficult to leave,” a veiled woman said in Qatar. “I love my country.”

The World Health Organization and UN children’s agency Unicef called for a humanitarian air bridge to deliver aid to Afghanistan to help more than 18 million people in need.

The White House said late Sunday that since last weekend, when the Taliban captured Kabul from the fleeing national government, the US “has evacuated and facilitated the evacuation of approximately 30,300 people on military and coalition flights”.

Up to 15,000 Americans need to be removed from Afghanistan, according to Biden, who says the administration wants to get at least 50,000 Afghan allies and their family members out of the country.

On one of the evacuation flights, an Afghan woman gave birth on board moments after landing at Ramstein Air Base in Germany on Saturday, the US Air Force said on Twitter. The woman delivered a baby girl in the cargo bay of a C-17 aircraft. The mother and baby were in good condition, the tweet said.

Additional reporting by Agence France-Presse and Bloomberg