The United States, which has thousands of troops trying to secure the airport, has set a deadline to complete the evacuations by Aug 31.
But there are up to 15,000 Americans and 50,000 to 60,000 Afghan allies who need to be evacuated, according to the Biden administration.
There are countless others who fear repression under the Taliban and are also trying to flee.
US President Joe Biden has described the evacuation operations as “one of the largest, most difficult airlifts in history”.
The situation was further complicated on Saturday when the US government warned its citizens to stay away from the airport because of “security threats”.
No specific reason was given, but a White House official later said Biden had been briefed on “counter terrorism” threats, including the Islamic State group.
European Union foreign policy chief Josep Borrell gave a bleak assessment of whether the airlifts would succeed.
“They want to evacuate 60,000 people between now and the end of this month. It’s mathematically impossible,” European Union foreign policy chief Josep Borrell told AFP.
Borrell added that “we have complained” to the Americans that their airport security was overly strict and hampering attempts by Afghans who worked for the Europeans to enter.
On Saturday, the Pentagon said 17,000 people had been taken out since the operation began on Aug 14, including 2,500 Americans.
Thousands more have left on other foreign military flights.
The Taliban have been publicly content to allow the US military oversee the airlift, while focusing on how they will run the country once the foreign forces leave.
Taliban co-founder Mullah Abdul Ghani Baradar flew into Kabul and planned to meet jihadi leaders, elders and politicians in the coming days, an official told AFP.
Among them are leaders of the Haqqani network, a US-designated terrorist organisation with million-dollar bounties on its leadership.
The Taliban stunned the world when they swept into Kabul last week, ending two decades of war, facing virtually no opposition from government forces that had been trained and equipped by the US-led alliance.
However there have been since been flickers of resistance, with some ex-government troops gathering in the Panjshir Valley, a mountainous region north of Kabul long known as anti-Taliban bastion.
Source: Channel News Asia