Taliban fighters have entered Kabul and sought the unconditional surrender of the central government, officials said.
The Afghan government, meanwhile, hoped for an interim administration, but increasingly had few cards to play – amid reports President Ashraf Ghani has left the country.
Taliban spokesman Suhail Shaheen told Qatar's Al-Jazeera English satellite news channel that the insurgents are "awaiting a peaceful transfer of Kabul city".
He declined to offer specifics on any possible negotiations between his forces and the government.
But when pressed on what kind of agreement the Taliban wanted, Shaheen acknowledged that they were seeking an unconditional surrender by the central government.
Also on Sunday, Afghanistan's acting Interior Minster Abdul Satta Mirzakwal said in a video: "The security of the city has been guaranteed. There will be no attack on the city.
"The big city and the power will be handed over to a temporary third party and it is our duty to secure the city.
"Don't be victim of their propaganda. Kabul will be safe. Don't worry, there won't be any problems in Kabul."
Negotiations in Doha, Qatar, the site of a Taliban office, have failed to stop the insurgents' advance.
Acting Defence Minister, General Bismillah Khan Mohammadi, has remained adamant the capital is "secure".
He said: "Honourable and faithful people of Kabul, I, as a representative of the security and defence forces of Afghanistan, assure you all that Kabul is secure, your security and defence forces are committed to their promise of defending Kabul.
"International forces are ready to provide whatever help to the Afghan security forces. As you all know, the president yesterday had a meeting with the senior leadership of the country, in that meeting he gave them full authority, so that from tomorrow they can go to Doha and reach an agreement on Afghanistan.
"Until there is an agreement in Afghanistan, in Kabul, I assure you that Kabul is secure, your children, your security forces, everyone is ready to defend Kabul.
"Just some enemies of Kabul are trying to spread propaganda in order to make people lose confidence. I, as a representative of the Afghan government, of the defence forces of Afghanistan, assure you that until there is an agreement Kabul will be secure, stay confident."
US forces have already arrived in Kabul to help evacuate some personnel – military helicopters shuttled between the US Embassy compound and the airport.
UK troops have already begun deploying on Operation Pitting, to assist with the evacuation of UK nationals and former British staff eligible for relocation under the Afghan Relocation and Assistance Policy (ARAP).
UK Foreign Secretary Dominic Raab wrote on Twitter that he and Shah Mahmood Qureshi, Foreign Minister of Pakistan, agreed "it is critical that the international community is united in telling the Taliban that the violence must end and human rights must be protected".