The defence secretary says the UK will leave Kabul alongside next week’s planned US withdrawal.
The UK has “hours now, not weeks” to evacuate people from Afghanistan, the defence secretary has said.
Ben Wallace said troops would leave Kabul’s airport when the US withdrew, which is due to happen on 31 August.
At an emergency meeting of G7 world leaders on Tuesday, Prime Minister Boris Johnson will ask US President Joe Biden for an extension of that deadline in order to allow more flights.
But a spokesman for the Taliban said it would not support a deadline extension.
Suhail Shaheen said an extension would mean extending Afghanistan’s occupation, and warned of consequences if that were to change.
Thousands of people are waiting to board flights at Kabul’s international airport, just over a week after the Taliban seized the capital.
The BBC’s chief international correspondent Lyse Doucet says she is receiving “desperate calls for help” on her phone “hour-by-hour and sometimes minute-by-minute” from those who know the window to get out is closing.
“Please my life is in danger. Please can you get me on a flight,” says one. Another reads: “Please I can’t get to the airport, can you get me inside?”
Armed forces minister James Heappey told the BBC that 6,631 people had been evacuated to the UK in the past week, and that nine flights were planned for the next 24-hour period.
The Taliban’s swift takeover of Afghanistan shocked its people and the world. It happened after foreign forces announced their withdrawal following a deal between the US and the Taliban, two decades after American-led forces removed the militants from power in 2001.
Mr Wallace said: “I don’t think there is any likelihood of staying on after the United States.
“We are really down to hours now, not weeks. We have to make sure we exploit every minute to get people out.”
Downing Street said on Monday evening that the prime minister had spoken to US President Joe Biden, before Tuesday’s G7 summit, saying they had “committed to driving international action” to “stabilise the situation”.
They also agreed to continue working together to ensure that people eligible to leave can do so “including after the initial phase of the evacuation has ended”.
The UK would continue its evacuation process “as long as the security situation allows”, No 10 said.
The prime minister’s official spokesman said “discussions on the ground” had been held with the Taliban over extending the deadline, but officials were still working towards 31 August.
The UK’s former ambassador to Afghanistan, Sir Nicholas Kay, told BBC Radio 4’s PM programme he believed there was “scope to engage with the Taliban” to agree on evacuations out of Kabul beyond 31 August.
He said there was an opportunity to “reframe the problem as to how to agree with the Taliban a plan for an organised operation, evacuation, relocation of these people”.
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Lloyd Comer, 60, served in the Army for 35 years but has worked in the private sector in Kabul since 2013.
He was told the Taliban was approaching his area, and contacted the Foreign Office, but he was told to stay put.
Mr Comer disregarded the advice – deciding to make a journey to the airport and choosing to forego an armoured vehicle, which he thought might become a target.
He told BBC Radio 4’s Today programme he dressed in Afghan style to disguise himself before being driven in the back of a white Toyota Corolla through Taliban checkpoints to Kabul airport.
He said those driving him “bundled me out through the crowds with my equipment and got me inside the gates of the Baron Hotel which is the secure site”.
He is now back home in Nottingham after leaving Kabul via the United Arab Emirates and a charter flight to Birmingham.
“The Foreign Office called me an hour ago to ask me if I was still in Kabul,” he said. “It was emotional, I’m home, I’m safe.”
“My heart bleeds for those people we’ve left behind,” he added. “The guilt that you feel for leaving your friends and those loyal staff when they probably needed me most is overwhelming.”
Other US allies including France, Germany and Italy have also said they need more time beyond the 31 August deadline.
Mr Heappey said about 1,800 “eligible people” or UK passport holders remained in Afghanistan – as well as 2,275 Afghans who can be resettled having worked for the UK government, and a further list of people from “wider Afghan civil society who we would like to get out if we’re able”.
He said airport checks were necessary for UK security and said there were “people right now in Kabul trying to get onto British flights that we have identified in our checks as being on the UK no-fly list”.
Mr Heappey said there was a “hard reality that we won’t be able to get out everybody that we want to”.
But he said the airlift was not the only route out of Afghanistan, adding there was a “second phase” to the resettlement programme planned for when the UK fully withdraws from Kabul.
Under the second phase, Afghans will be able to have resettlement claims processed at refugee camps or UK embassies in countries neighbouring Afghanistan.
Behind the scenes Britain has been pushing the US for an extension of the Kabul airlift.
Not for long – just a few days. That short time would allow evacuation flights to continue for most of this week, while giving the military a few extra days next week to pack up and leave.
Military planners in the Ministry of Defence estimate they need between 24 and 36 hours to fold up the operation and get every soldier out.
Hopes of an extension still appear slim – not least because of the Taliban.
Some have asked why the UK and allies could not stay longer without the US? But the US is providing the bulk of the troops to keep the airport secure. It’s also essentially running the airport – including air traffic control.
The US is also providing intelligence and surveillance for the operation. Replacing these elements would take significant time and resources. Without US military power, there’d also be an increased risk for those left behind.
Staying in Afghanistan without the US was never really an option for the British. Nor is it at Kabul airport.
Mr Heappey added that the Taliban had been an “effective partner” in the evacuation – but the UK was “taking nothing for granted” with the militants.
Labour leader Sir Keir Starmer has written to the prime minister calling for more information on how the UK is planning for the mission’s next stages.
Sir Keir asked Mr Johnson whether he had spoken to Mr Biden to request an extension to the evacuation period and whether the UK was working on a contingency plan with Nato allies to “hold Kabul airport without US troops”.
The Foreign Office said it had sent five additional members of staff to Kabul to help with the evacuation, bringing the total number of its staff there to 19.
Additional reporting by George Bowden