Joe Biden confirmed the US was on track to withdraw troops by August 31, in a White House press conference that ran hours late.
Prime Minister Boris Johnson earlier claimed the US president had refused to keep US troops in Afghanistan after the evacuation deadline.
Speaking after an emergency meeting of the G7 earlier, Mr Johnson revealed the President had denied his request to keep American troops at Kabul airport.
He said allied forces would "go on right up to the last moment that we can", and "we're confident we can get thousands more out".
But he added: "The situation at the airport is not getting any better, there are public order issues, there are harrowing scenes for those who are trying to get out, and it's tough for our military as well."
Mr Biden confirmed the US is on track to complete its evacuation plan to withdraw troops from Afghanistan by August 31, in Tuesday's press conference following hours of speculation.
But although troops were on pace to finish by the date, the president added he had asked the Pentagon to prepare "contingency" plans if necessary.
“The sooner we can finish the better. Each day of operations brings added risk to our troops,” Mr Biden said.
He told reporters on Tuesday troops, allied forces, and civilians on the ground at the airport faced a threat from Taliban enemy and terror group affiliate ISIS-K as its operatives targeted the site.
He said the ongoing operation depended on the cooperation of the Taliban to allow the US access to complete its mission.
It is understood the President earlier addressed a virtual meeting of the G7, which included the PM, French President Emmanuel Macron and German Chancellor Angela Merkel, for around seven minutes but refused to back down.
Confirming August 31 remained the deadline, Pentagon spokesman John Kirby said: “We’re absolutely still aiming towards the end of the month.”
G7 leaders issued a joint statement saying: “We express our grave concern about the situation in Afghanistan and call for calm and restraint to ensure the safety and security of vulnerable Afghan and international citizens, and the prevention of a humanitarian crisis.”
Speaking to broadcasters after the meeting, the PM said there were “harrowing scenes” at Kabul airport for those attempting to flee Afghanistan.
“We have evacuated 9,000, the UK alone has taken 9,000 people out of Kabul, I think 57 flights – huge, huge effort by our military,” he said.
“We will go on right up until the last moment that we can. But you have heard what the President of the United States has had to say, you have heard what the Taliban have said.
“I think you have got to understand the context in which we’re doing this, we’re confident we can get thousands more out."
More than 9,000 people had been flown out of Kabul on 57 RAF flights in what Mr Johnson branded “an extraordinary airlift”.
Mr Biden had been roundly criticised before the meeting by MPs from all parties America's chaotic withdrawal from the war-torn country.
The Taliban seized control of Afghanistan last Sunday, with even Mr Biden admitting militants' lightning rise to power was quicker than western leaders predicted.
It came as Taliban spokesman Zabihullah Mujahid told Afghans to stop travelling to Kabul airport as no evacuations would be permitted after the August 31 deadline.
During a press conference in Kabul, he said: “August 31 is the time given and after that it’s something that is against the agreement.
“All people should be removed prior to that date.
“After that we do not allow them, it will not be allowed in our country, we will take a different stance.”
A joint statement from G7 leaders, meanwhile, said they had agreed a "renewed humanitarian effort by the international community".
It reads: “To this end we support the UN in coordinating the immediate international humanitarian response in the region, including unfettered humanitarian access in Afghanistan, and will contribute collectively to that response.
“As part of that, we will cooperate together and with neighbouring and other countries in the region on supporting Afghan refugees and host communities as part of a coordinated long-term regional response.
“We call on all partners of Afghanistan to support this effort and wider regional stability through multilateral channels.”
The PM added: “The number one condition we’re setting as G7 is that they’ve got to guarantee right the way through, through August 31 and beyond safe passage for those who want to come out.
“Some of them will say that they don’t accept that, some of them I hope will see the sense of that, because the G7 has very considerable leverage, economic, diplomatic and political.” Challenged that his diplomatic efforts had failed, Mr Johnson claimed the G7 had “huge leverage” when engaging with the Taliban.
“I am totally realistic about the Taliban and I don’t think that anybody is going to pretend that this is anything other than a very difficult situation,” the PM said. “But that doesn’t mean that we should ignore the leverage that we have.
“We want to help with the humanitarian crisis, the difficulties that people in Afghanistan, people fleeing Afghanistan, are going to experience.”
Shadow Foreign Secretary Lisa Nandy mocked the PM’s “Global Britain” slogan and called the failure to change Mr Biden’s mind “shameful”. She said: “Even at this eleventh hour we hoped the PM would make good on that commitment and bring world leaders together to agree a joint approach around an extension.
“Those efforts have fallen short and the painful reality is that there will be many people – Britons and the Afghans who supported us over two decades – who simply aren’t going to get out. This is a really dark moment for many Afghans.”
She added: “It is a very shameful moment for the Government that after 18 months of warning and a failure to plan we have now had this failure at the G7. We’ve got to confront the reality now that every hour counts.”
Earlier, gloomy Defence Secretary Ben Wallace insisted British troops would not try to secure the airport after the US pullout.
The former Scots Guards captain said|: “It’s not about effectively whether I could fly in thousands of troops and secure the airport. Yes I could do that, I could probably secure the airport for a few months, or maybe a year or two.
“But for what purpose? For them to be shot at, attacked, people not to get to the airport and to trigger a permanent fight? I don’t think that is a solution.”
