Watch: Boris Johnson discusses 31 August Afghanistan deadline
It follows emergency talks at a virtual G7 summit hosted by Johnson on Tuesday afternoon.
The prime minister had hoped to convince the president to keep the remaining troops in Kabul to allow for more evacuations following the Taliban's seizure of Afghanistan earlier this month.
Biden, however, is pressing ahead with his withdrawal plans.
The Taliban had warned on Monday there will be "consequences" if foreign soldiers do not leave the country by the 31 August deadline.
It then stepped up the pressure ahead of the summit on Tuesday by saying no further evacuations will be allowed after this date.
In a broadcast clip after the meeting, however, the PM said the UK and the other G7 nations remain committed to the "safe passage" of evacuees after 31 August.
He said: "The number one condition we’re setting as G7 is that they [the Taliban] have got to guarantee, right the way through, through 31 August and beyond, safe passage for those who want to come out."
Johnson did not explain how this would work without US military support, with US soldiers having provided security at Kabul airport to allow people to flee the country.
He only said the G7 – a group which includes the US – "has very considerable leverage: economic, diplomatic and political".
He said this demand is part of a "road map for the way in which we’re going to engage with the Taliban".
Johnson was also asked if he made clear his "frustration" to Biden – who spoke at the G7 summit for about seven minutes – about the president's refusal to extend the deadline, but he refused to answer.
He responded: "Let's be clear, the immediate phase of the evacuation has actually been a very considerable success by the military."
Why did it all come down to 31 August in the first place?
It's a continuation of the February 2020 "peace" agreement struck between Biden's predecessor Donald Trump and the Taliban to end US involvement in Afghanistan.
Earlier this year, Biden followed through on Trump's agreement.
On 14 April, Biden pledged: "US troops, as well as forces deployed by our NATO Allies and operational partners, will be out of Afghanistan before we mark the 20th anniversary of that heinous attack on 11 September, 2001."
"It’s time to end the forever war," the president said.
On 8 July, with troops already beginning to depart, Biden then announced: "Our military mission in Afghanistan will conclude on 31 August."
Biden's commitment, of course, was complicated by the Taliban's stunning seizure of Afghanistan this month.
The president is seen by many to have acted unilaterally with his decision to withdraw troops, in turn leaving allies such as the UK exposed.
On the same day as Biden's announcement last month, Boris Johnson confirmed that "most of our personnel have already left".
The subsequent debacle in Afghanistan has hugely damaged Biden's international standing, with Tony Blair, who sent UK troops to Afghanistan 20 years ago following the 9/11 attacks, saying the withdrawal was "in obedience to an imbecilic political slogan about ending ‘the forever wars’".
He added Britain has a “moral obligation” to stay until “all those who need to be are evacuated”.
On Monday, the situation was further complicated as a Taliban spokesman warned there will be "consequences" if foreign soldiers do not leave Afghanistan by Biden's agreed deadline of 31 August.
Suhail Shaheen said extending evacuations beyond that date would be considered a "red line" and would “provoke a reaction”.
He told Sky News: "If the US or UK were to seek additional time to continue evacuations – the answer is no."
On Tuesday, another Taliban spokesman, Zabihullah Mujahid, said no evacuations would be permitted after the deadline as he urged Afghans to “return to their homes and resume their calm everyday lives”.
Watch: Joe Biden defends Afghanistan withdrawal (from Monday)