The US defence department ordered the emergency use of commercial aircraft to assist in Kabul evacuations.
The 18 aircraft from American Airlines, Atlas Air, Delta Air Lines, Omni Air, Hawaiian Airlines, and United, will not be flown into Kabul but will instead be used to move people from "temporary safe havens" and military bases.
Western countries are rushing to evacuate their citizens in addition to Afghans who helped the NATO effort and others who are vulnerable after the Taliban took over the capital city a week ago.
At least seven people died in the chaos outside the Kabul airport, the British military said on Sunday, as people desperately flood the airport trying to flee.
"Conditions on the ground remain extremely challenging but we are doing everything we can to manage the situation as safely and securely as possible,” a British ministry of defence spokesperson said in a statement.
The US embassy in Kabul meanwhile warned that there were "potential security threats" outside the airport and advised Americans "to avoid traveling to the airport and to avoid airport gates at this time".
US President Joe Biden has said the airport in Kabul was secure, with thousands of troops facilitating evacuations from the country. Biden had pledged to evacuate Afghans who helped the United States including translators and others who assisted in the war effort.
Videos have depicted chaos outside the airport in Kabul as those who pass the Taliban checkpoints make it to the area where evacuations are being carried out.
Evacuees begin to arrive in EU countries
Britain's ambassador to Afghanistan, Laurie Bristow, said in a video statement that the country had managed to evacuate 5,000 people so far. He said 1,000 people had been evacuated in the past 14 hours.
Germany's federal defence forces said on Sunday that they have so far evacuated 2,100 people from 38 nations.
They said the situation continued to be very difficult with airport gates needing to be closed due to crowds. The forces said they will continue to evacuate for as long as possible.
Italy said its military evacuated nearly 1,000 Afghan citizens out of Kabul over the past week, with two flights carrying 207 Afghans arriving in Rome on Saturday, the Associated Press reported.
The Dutch defence ministry said a total of 368 evacuees with a Dutch or foreign passport have now arrived in the Netherlands.
The Netherlands said they would work to evacuate interpreters, security guards, legal assistants, cooks and drivers as well as employees of Dutch development projects, human rights defenders and, in particular, women's rights defenders.
Journalists and fixers working for Dutch media area also concerned, the defence ministry said.
French armed forces minister Florence Parly said last week that France had evacuated 570 people from Kabul.
Europe fears new migration crisis
As evacuations of Afghans who worked with EU countries continued, interior ministers discussed concerns that the chaos could lead to a new migration crisis.
EU home affairs commissioner Ylva Johansson said in a statement the bloc "should not wait until people arrive at the external borders".
"We should prevent people from heading towards the European Union through unsafe, irregular and uncontrolled routes run by smugglers," Johansson added after the meeting of interior ministers.
In a televised speech earlier in the week, French President Emmanuel Macron said Europe needed to anticipate and protect against "significant irregular migratory flows."
Many officials are worried about a repeat of the 2015 migrant crisis, when a record 1.25 million first-time asylum applicants were registered in the European Union.
That crisis was fuelled in part by the Syrian civil war.