Boris Johnson's special representative for Afghan transition has travelled to Qatar to meet with "senior Taliban representatives", Downing Street confirmed.
Sir Simon Gass is said to have made the trip to stress the importance of allowing people to leave Afghanistan.
Home Office minister Victoria Atkins said the government was unable to confirm the exact number of people left there who are eligible to live in the UK, but said "we know there are a few hundred UK nationals still left in Afghanistan".
"We want to try get those people out as safely and as quickly as we can," she said.
Those still stranded there have been told they could be processed for resettlement in the UK if they can access neighbouring countries via land borders.
Ms Atkins said discussions with third countries are underway to "ensure that people who reach borders are processed quickly and safely".
"But we also need to talk to the Taliban," she added.
Asked if Britain can trust the Islamist group, the minister said: "We will judge the Taliban by their actions. I am pleased to hear some of the assurances that have been given this morning and in recent times but of course actions will count more than words."
The news emerged as the Foreign, Commonwealth and Development Office (FCDO) announced 15 crisis response specialists are being deployed to Pakistan, Uzbekistan and Tajikistan to assist British diplomats in their work helping people escape Taliban-controlled Afghanistan.
As the final US troops left Kabul, so did the hopes of many that they could be flown out of Afghanistan - leading many families to escape the country by entering Pakistan.
Pictures showed dozens of Afghans receiving food from a Christian organisation on the outskirts of Chaman, a border town in the Pakistan's southwestern Baluchistan province - a day after the US wrapped up its 20-year military presence in the country.
The officials are expected to arrive within the next 48 hours, with the focus on helping UK nationals, interpreters and other Afghans who were employed by the UK, and those Afghans judged most at risk.
Foreign Secretary Dominic Raab said the deployment of the three teams would bolster efforts to help people cross the border.
On Tuesday, Mr Raab said the number of UK nationals left behind in Afghanistan is in the “low hundreds”.
He was unable to give a “definitive” figure on how many Afghans the UK had failed to airlift to safety after the Taliban seized power as the US withdrew its final troops following a 20-year mission.
In a statement on Wednesday, Mr Raab said "we know not everyone who wanted to leave and were eligible for UK support could.
“We will stand by them, and we’re working with partners in neighbouring countries to support onward travel to the UK.
“These latest rapid deployment teams will bolster those efforts and reinforce our embassy teams on the ground to help those in need.”
Elsewhere, the Home Office said Afghans who worked with the British government and military will be able to move to the UK permanently.
Former Afghan staff and their family members eligible for the Afghan Relocations and Assistance Policy (Arap), which prioritises relocation to the UK for current or former locally employed staff who have been assessed to be under serious threat to life, will be given immediate indefinite leave to remain as opposed to only five years’ temporary residency as previously permitted.
It is part of the government’s Operation Warm Welcome, which is designed to help Afghans rebuild their lives in the UK.
More than 8,000 Arap claimants were among the 15,000-plus people evacuated by the UK since August 13.
Ms Atkins said the scheme was announced for the thousands of Afghans who have worked with British forces over the past 20 years - "we want to help them because they have helped us", she said.
"We now want to give those people a very warm welcome and help them build bright lives in our country."
Meanwhile, MI6 chief Richard Moore held talks with the Pakistan military on Afghanistan in recent days, according to media reports in Pakistan.
The Daily Telegraph also reported secret discussions have taken place between senior British intelligence officials and the Taliban in Kabul in the past fortnight.
The newspaper said the aim of the talks was to seek assurances that Afghanistan will not be used to launch terrorist attacks on the west.
Mr Raab admitted to ITV News on Tuesday that the UK's safety is uncertain following the withdrawal of Western forces from Afghanistan.
Mr Raab said: "I think it's very difficult to judge at this stage, it's very fluid, obviously we're concerned. We've seen the ISIS-K attack on Abbey Gate by the airport."
But, he added, "in relation to military matters, we retain the right to exercise self-defence, including against terror groups".
Mr Raab will also face MPs on Wednesday to discuss the government’s handling of the Afghanistan crisis.
With doubts about his future as Foreign Secretary, No 10 insisted Prime Minister Boris Johnson has “full confidence” in Mr Raab.
Some of the ire was for remaining in holiday in Crete as the Taliban was seizing back control of Afghanistan.
Mr Raab will be quizzed by the Commons Foreign Affairs Committee later on Wednesday over his response to the crisis in Afghanistan - for which many have blamed the US and UK.
Asked whether she thinks the foreign secretary is under pressure, Ms Atkins said Mr Raab's "personal commitment to this is genuine".
Shadow foreign secretary Lisa Nandy insisted her counterpart has questions to answer on “much more than the chaos of the last two weeks”.
She said: “This has been the biggest foreign policy failing in a generation.
“The Foreign Secretary has serious questions to answer when he appears before the Foreign Affairs Committee.”
Ms Nandy added: “The Foreign Secretary had 18 months to prepare but was missing in action.
“As a result, on his watch Britain has become weaker in the world and faces greater risks from terrorism.”