Boris Johnson will personally lobby Joe Biden at the G7 leaders summit, No 10 said on Sunday night, pleading with him to keep US troops at Kabul airport beyond the end of August, after a weekend of tension between the UK and its closest ally.

With the Taliban tightening their grip at the airport, Downing Street confirmed the PM would lobby the US president on Tuesday at the virtual meeting. The news came as Taliban forces sought to assert their authority, blaming the US for the “anarchy” at the airport and insisting they were the only ones capable of restoring order. Throughout the day, Taliban fighters fired their guns into the air and used batons to force people into orderly queues in an attempt to halt the desperate scenes in which at least 20 people have died.

No 10 confirmed on Sunday that the prime minister would ask Biden to maintain a presence after 31 August. It is feared thousands of people with British connections will miss the chance to be evacuated as the Taliban begin to partly administer evacuations. At a press conference on Sunday night, Biden suggested that the date for ending evacuations could be pushed back, saying discussions were ongoing, but added: “Our hope is we will not have to extend.”

A British minister said the flow of people outside the airport had improved as a result of the Taliban’s intervention, allowing the pace of the UK evacuation to increase. The Times reported on Sunday night that RAF had already extended its own evacuation deadline. It was planned to end on Tuesday but would now be Friday or Saturday, the paper said, with more people, including Afghan politicians and humanitarian workers, eligible for evacuation.

James Heappey, the armed forces minister, said the Taliban were “marshalling people into separate queues for the US evacuation and the UK evacuation, and that is making a big difference to the size of the crowds outside the UK gate and allowing us to process people much more quickly”.

The Taliban accused the US, which is holding the airport with 5,200 troops, of failing to keep control properly – though it is the group’s fighters who have beaten and shot at those trying to access the site. At least 20 people have died in the chaotic scenes both on the tarmac and outside the airport since the previous Sunday, a Nato official told Reuters. Many other Afghans have been held at roadblocks or are too scared to travel despite receiving an evacuation offer.

Amir Khan Muttaqi, the chief of the Taliban’s guidance council, said: “All Afghanistan is secure, but the airport which is managed by the Americans has anarchy,. The US should not defame itself, should not embarrass itself to the world and should not give this mentality to our people that [the Taliban] are a kind of enemy.” In his press conference Biden said that the US had expanded their perimeter around the airport in order to accelerate their evacuations. He said the Taliban had cooperated with those efforts.

It remains unclear how long the US will remain at the airport to continue the airlift, although it emerged that Britain had previously been desperately lobbying the Biden administration for it to consider staying on past a previously announced deadline of 31 August – to no avail.

Heappey said that the foreign secretary, Dominic Raab, who is under fire for his own handling of the crisis, had “made representations” to his counterpart, Antony Blinken, the US secretary of state, on the issue. But the British had received no reassurances, prompting Heappey to say in a TV interview that “we are assuming nothing” when it came to a final exit date.

Just over 1,900 people were evacuated as part of the RAF operation in the 24 hours to Sunday night, the largest one-day total, taking the overall figure to 5,725 since 12 August.

The Labour leader, Sir Keir Starmer, wrote to the prime minister asking a string of questions given “the lack of foresight and planning” over Afghanistan, including if there was a joint evacuation plan made by UK and US commanders in Kabul.

Other developments on Sunday included:

Up to 300 Afghan former employees of the British security firm G4S say they are stranded in Kabul in fear of their lives and feel abandoned by the company and the UK government.

Tony Blair, the prime minister who authorised the UK contribution to the US-led invasion of Afghanistan, warned the chaotic withdrawal would provide opportunities for terror groups and countries hostile to the west.

One of the few remaining rebel leaders opposed to the Taliban, Ahmad Massoud, warned a new civil war was “unavoidable” unless there was a power-sharing agreement.

Britain said it would establish offshore asylum centres for Afghan refugees in countries such as Pakistan and Turkey, as ministers admitted the UK could not rescue all those eligible for resettlement in time. But on Sunday night Turkey said it would reject any request to set up a centre from the UK.

Officials confirmed Johnson would use the G7 meeting to push Biden over the US extending its presence in Kabul, but stressed the event was primarily about long-term solutions. On Sunday night, Reuters reported that Britain believes the G7 should consider economic sanctions and withhold aid if the Taliban commits human rights abuses and allows its territory to be used as a haven for militants. Asked about sanctions at his press conference, Biden said he could support them “depending on context”.

In his letter, Starmer asked Johnson if he had spoken personally to Biden to seek an extension to the US presence. Johnson and Biden last spoke on Tuesday, a call that took over 24 hours to arrange, according to the Sunday Times. Biden did not name the UK on Sunday night when listing the countries whose leaders he had spoken to.

Other issues raised in Starmer’s letter included whether there were discussions over Nato allies temporarily holding Kabul airport without US troops; whether Nato forces could ensure safe areas to provide airport access; and if there was scope for a UN operation to assist.

Lord Ricketts, a former national security adviser, said Biden’s decision to withdraw from Afghanistan and his approach to handling the retreat was “a wake-up call to allies who had nourished hopes of a return to US internationalism”.

“From what I hear the US is totally focused on avoiding a disaster befalling the evacuation and they don’t have time for any other allies at the moment,” he said. “But the lack of communication is something the Brits feel most: we are most needy about wanting to be seen as Washington’s closest counterparts.”

No 10 deflected questions about disagreements with the US, with officials stressing the “excellent working relationship” between the leaders.

In parallel, there was renewed focus on the absence on holiday of Raab as Kabul fell to the Taliban, after a report that the foreign secretary had pressured Johnson into extending his absence. Raab had been instructed to return on 13 August, as the Afghan government faced collapse, but reportedly “nobbled” the prime minister into allowing him to remain on holiday in Crete for two extra days.