The UK is seeking international agreement to ensure the Taliban sticks to its commitment to allow safe passage for Afghans and foreign nationals who want to leave Afghanistan.
A series of diplomatic meetings will be held this week with Foreign Secretary Dominic Raab set to hold talks with Turkish and Qatari officials.
It comes as troops arrived back in the UK after ending their Afghan operation.
The government says it will continue to help people eligible for resettlement.
While more than 15,000 people have been evacuated by the UK since 14 August, hundreds of people who are entitled to come to the country have been left behind.
It is feared that about 800 to 1,100 eligible Afghans, including those who worked for the UK government, and 100 to 150 British were unable to get on evacuation flights.
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The Foreign Office says the UK and many other countries had been given assurances by the Taliban that foreign nationals and Afghan citizens with proper documentation will be allowed to leave the country.
Even if the Taliban is true to its word there is still uncertainty for those who are able to get to its borders as processing centres have yet to be set up in neighbouring countries.
During the series of meetings to be held on Monday officials will try to rally the international community to make sure the group keeps its promises.
Mr Raab will use a US-chaired virtual meeting to talk to representatives from Turkey and Qatar, who are considered to have more influence over the militant group than Western nations.
During the discussions, which will also include G7 counterparts and Nato, the foreign secretary is expected to underline that Afghanistan must not become a haven for terrorists, with stability in the region a priority, and will underline the need to hold the Taliban to account over human rights promises.
A meeting of the ambassadors from the five permanent members of the UN Security Council – the US, France, China, Russia and the UK – will also take place.
On Sunday Boris Johnson said the UK and its allies would “engage with the Taliban not on the basis of what they say but what they do”.
“If the new regime in Kabul wants diplomatic recognition, or to unlock the billions that are currently frozen, they will have to ensure safe passage for those who wish to leave the country, to respect the rights of women and girls, to prevent Afghanistan from, again, becoming an incubator for global terror, because that would be disastrous for Afghanistan,” he said.
Labour has criticised the government’s handling of the crisis and accused ministers of being “missing in action”.
Shadow foreign secretary Lisa Nandy has written to Mr Raab warning him that the government was working with a “serious underestimate” of the number of people eligible for evacuation who had been left behind.
The BBC’s Lyse Doucet, in Kabul, says she and her colleagues are still receiving urgent SOS messages from Afghans who feel threatened by the Taliban.
They include musicians, university students and female politicians, and many say the Taliban are stopping them from leaving.
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What happens to Afghan refugees coming to the UK?
Arrivals on official flights enter a 10-day Covid quarantine in a hotel
Government officials and local authorities are trying to find them permanent homes
A shortage of suitable accommodation means many will be placed in hotels
Some will get refugee status and can live in the UK permanently
Others will get a five-year visa to live and work in the UK – and can then apply for permanent residence
Afghans arriving independently will enter the normal system for asylum claims – which has a backlog of 70,000 people
These people cannot settle, or work, while their claims are considered