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At an emergency meeting of G7 leaders Boris Johnson was expected to press the US President to keep his forces on the ground at Kabul airport to continue providing security.
But following the crisis talks on Tuesday afternoon, Mr Biden said the deadline remained in place.
The US president’s press secretary Jen Psaki said withdrawing by August 31 would see the war end “based on the achievement of US objectives”.
Ms Psaki said: “During a meeting this morning with the G7 leaders, the president conveyed that our mission in Kabul will end based on the achievement of our objectives.
“He confirmed we are currently on pace to finish by August 31 and provided an update on progress in evacuating Americans who want to come home, third-country nationals, and Afghans who were our allies during the war.
“He also made clear that with each day of operations on the ground, we have added risk to our troops with increasing threats from Isis-K.
“In addition, the president has asked the Pentagon and the State Department for contingency plans to adjust the timeline should that become necessary.”
Earlier on Tuesday, Pentagon spokesman John Kirby insisted the US was committed to evacuating the Afghan commanders and other troops currently helping with the evacuation.
“There’s been no change to the timeline of the mission which is to have this completed by the end of the month,” he added.
It comes as the Taliban spokesman Zabihullah Mujahid said his group will accept “no extensions” to the evacuation deadline for the withdrawal of all American troops.
“August 31 is the time given and after that it’s something that is against the agreement,” Mr Mujahid told a press conference in Kabul.
Mr Johnson said the UK would continue with evacuations “until the last minute that we can”.
“But you’ve heard what the President of the United States has said, you’ve heard what the Taliban have said,” he added.
“We’re confident we can get thousands more out, but the situation at the airport is not getting any better.”
Mr Johnson said there were “harrowing scenes” at Kabul airport but the G7 group of nations had agreed a “roadmap” for future engagement with the Taliban, which included insistence people who want to leave Afghanistan beyond the deadline have “safe passage”.
“We have got together the leading Western powers and agreed not just a joint approach to dealing with the evacuation, but also a road map for the way in which we’re going to engage with the Taliban,” he said.
“The number one condition we’re setting as G7 is that they have got to guarantee, right the way through, through August 31 and beyond, safe passage for those who want to come out.
“Some will say that they don’t accept that and some, I hope, will see the sense of that, because the G7 has very considerable leverage – economic, diplomatic and political.”
Following the meeting he said Afghanistan cannot “lurch back into becoming a breeding ground of terror” or “a narco state” where girls are not allowed to attend school.
“Those are important things that we value as G7, those are things that unite us in the West, those are things for which we fought for years in Afghanistan, and for which people in this country gave their lives,” he added said.
“The point that was made today by G7 leaders is that we remain committed to those values and we remain committed to Afghanistan.
“But the number one condition that we are insisting upon is safe passage beyond the [August 31], beyond this initial phase, for those who want to leave Afghanistan.”
Earlier, Taliban spokesman Mr Mujahid insisted life is returning to normal in Afghanistan and freedom of the media, now in operation, is improving.
Chaos at the airport remains a problem with many Afghans still desperate to flee the Taliban takeover.
The Taliban spokesman said the crowds of Afghans seeking a flight out of the country could go home and there was no list of people targeted for reprisals.
Responding to the announcement that the Kabul evacuation will not be extended, Labour’s shadow foreign secretary Lisa Nandy said: “This is a dark moment for the UK Government and for Afghans.
“The Prime Minister has failed to persuade President Biden to agree to an extension to evacuation efforts and the painful reality is that people will be left behind - that’s appalling and unconscionable.
“With 18 months to prepare for this, we are left with a desperate scramble, with heroic soldiers and diplomats on the ground trying to move mountains while the clock keeps ticking.”
Ms Nandy added the “urgent priority remains the evacuation” and said “all possible resource” should be focused “on getting as many people out as safely as possible”.