Boris Johnson will make the request at an emergency meeting of world leaders from the G7 countries on Tuesday.
Many thousands of people are crowded outside Kabul airport, desperate to leave after the Taliban swept to power.
The US intends to pull all troops out by the end of the month.
Mr Biden said discussions were taking place about extending the deadline, but that “our hope is we will not have to”.
He said US troops had expanded the perimeter around Hamid Karzai International Airport as part of efforts to speed up evacuation efforts, and the Taliban had cooperated with the move.
While UK government officials have said there is “no fixed date” for UK withdrawal, it is feared that without US troops on the ground the remaining allied forces will be unable to secure the area.
The UK has more than 1,000 Armed Forces personnel deployed in Kabul.
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The talks between the G7 group of countries, of which the UK is currently the chair, will take place virtually and it is understood the main focus will be the long-term future of Afghanistan.
But the ongoing evacuation will also be discussed.
At least 20 people have died outside the airport, a Nato official told Reuters news agency, and there are reports that some people were crushed to death.
The UK is also exploring how hubs could be set up in countries bordering Afghanistan to help those who have worked for British forces to reach the UK once flights out of Kabul are no longer possible.
Mr Johnson has said it is vital the international community works together to ensure safe evacuations, prevent a humanitarian crisis and support the Afghan people to “secure the gains of the last 20 years”.