A series of urgent terror warnings have rattled evacuation efforts overseen by US forces, who have been forced into closer security cooperation with the Taliban to prevent a repeat of Thursday's carnage at one of the facility's main access gates.
Scores of Afghan civilians were killed in the bombing claimed by the regional Islamic State-Khorasan group, along with 13 US troops -- several of them just 20 years old, the same length of time as US military operations in Afghanistan.
But Biden said Saturday that his military commanders believed a fresh attack could come "in the next 24-36 hours", calling the situation "extremely dangerous".
"I directed them to take every possible measure to prioritize force protection," he said after a briefing from his national security team.
Earlier, the Pentagon said it had killed two "high profile" targets -- logistics experts for the jihadist group -- and wounded another in the drone strike in eastern Afghanistan.
No civilians were hurt in the attack, Major General Hank Taylor told a news conference in Washington.
"The fact that two of these individuals are no longer walking on the face of the Earth, that's a good thing," Pentagon spokesman John Kirby said.
In recent years, the Islamic State's Afghanistan-Pakistan chapter has been responsible for some of the deadliest attacks in those countries -- massacring civilians at mosques, shrines, public squares and even hospitals.
Two senior health officials from the former Afghan administration told AFP the death toll from Thursday's blast had topped 100, including the 13 Americans. Some media have put the toll as high as 170.
The US embassy in Kabul has repeatedly told American citizens to leave areas around the main gates in recent days, as tensions rise before Tuesday's withdrawal deadline.
Gone are the crowds of thousands around the airport's perimeter, hoping to be let through and allowed onto a plane.
Now, the Taliban have sealed off roads leading to the airport and are only letting sanctioned buses pass.
AFP saw more than a dozen small- and medium-sized buses disgorge tense-looking passengers at the main gate of the airport on Saturday.
"We have lists from the Americans... if your name is on the list, you can come through," one Taliban official told AFP near the civilian passenger terminal.
At the airport on Saturday, heavily armed Taliban fighters were seen throughout the grounds and auxiliary buildings of the complex, while US Marines peered at them from the passenger terminal roof.
After a 20-year war, the foes were within open sight of each other, separated by just 30 metres (yards) -- and holding fire.
Also in view of the American troops were Badri special forces in humvees gifted to the Afghan defence forces, but now flying the white Taliban flag.
Taliban spokesman Bilal Karimi tweeted that the group's fighters had already moved into parts of the military side of Kabul airport, but the Pentagon has repeatedly stressed that US forces retained control over the gates and airlift operations.
Kirby however did confirm Saturday that the United States has begun withdrawing its military personnel from the airport, without saying how many had left.
With the Taliban poised to take over when the last US plane leaves, discussions have begun on resuming normal flight operations.
Turkish officials have held initial talks with the Taliban in Kabul about helping get the airport back up and running.
President Recep Tayyip Erdogan said the Taliban wants to oversee security at the airport, while Ankara runs the logistics.
French President Emmanuel Macron said Saturday that talks were in progress with the Taliban in Qatar to "protect and repatriate" at-risk Afghans and may involve "airlift operations".
He said France was counting on help from Doha -- which has good contacts with the Taliban, having hosted peace talks -- to organise further flights.
Under enormous criticism at home and abroad for his handling of the Afghan crisis and the US military withdrawal, Biden has pledged to stick to the airlift deadline and punish those responsible for the suicide blast.
"This strike was not the last," he said Saturday after the drone attack.
About 112,000 people have been flown out of the country since August 14, the day before the Taliban swept to power, according to the US government.
Many Western allies have announced an end to their airlifts, with some admitting that at-risk Afghans eligible for evacuation had been left behind.
The United Nations said it was bracing for a "worst-case scenario" of up to half a million more refugees from Afghanistan by the end of 2021.
The Taliban have promised a softer brand of rule compared with their first stint in power from 1996 to 2001, which ended when the United States invaded Afghanistan following the September 11, 2001 attacks.
But many Afghans fear a repeat of their brutal interpretation of Islamic law, as well as reprisals against those working with foreign militaries, Western missions or the previous US-backed government.