Taliban fighters entered the outskirts of Kabul on Sunday and sought the unconditional surrender of the central government, officials said.
Panicked workers fled government offices and helicopters began landing at the US Embassy in the Afghan capital evacuating personnel.
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.
Taliban negotiators are headed to the presidential palace Sunday to discuss the transfer, said an Afghan official who spoke on condition of anonymity for fear of reprisals. It remained unclear when that transfer would take place.
Three Afghan officials told The Associated Press that the Taliban were in the districts of Kalakan, Qarabagh and Paghman in the capital.
The militants later pledged not to take Kabul "by force" as sporadic gunfire could be heard in the capital.
"No one's life, property and dignity will be harmed and the lives of the citizens of Kabul will not be at risk," the Taliban said.
A Taliban spokesman said "we are awaiting a peaceful transfer of Kabul city", AP reported.
An Afghan official says troops have surrendered Bagram Airbase to the Taliban. The base is home to a prison housing 5,000 inmates.
Taliban negotiators were heading to presidential palace to prepare for a "transfer" of power, AP reported.
The insurgents issued the statement on Sunday as their fighters entered the outskirts of Kabul.
The militants themselves didn't acknowledge the advance, though they earlier took Jalalabad, near a major border crossing with Pakistan, the last major city other than Kabul not under their control.
In a nationwide offensive that has taken just over a week, the Taliban has defeated, co-opted or sent Afghan security forces fleeing from wide swathes of the country, even with some air support by the US military.
The rapid shuttle-run flights of Boeing CH-47 Chinook helicopters near the embassy began a few hours later as diplomatic armoured SUVs could be seen leaving the area around the post.
The US State Department did not immediately respond to questions about the movements.
However, wisps of smoke could be seen near the embassy's roof as diplomats urgently destroyed sensitive documents, according to two American military officials who spoke on condition of anonymity because they were not authorized to discuss the situation.
Sikorsky UH-60 Black Hawk helicopters, which typically carry armed troops, later landed near the embassy as well.
The Czech Republic also approved a plan to begin withdrawing their Afghan staff from their embassy after earlier taking their diplomats to Kabul International Airport.
President Ashraf Ghani, who spoke to the nation Saturday for the first time since the offensive began, appears increasingly isolated as well.
Warlords he negotiated with just days earlier have surrendered to the Taliban or fled, leaving Ghani without a military option. Ongoing negotiations in Qatar, the site of a Taliban office, also have failed to stop the insurgents' advance.
Thousands of civilians now live in parks and open spaces in Kabul itself, fearing the future. Some ATMs stopped distributing cash as hundreds gathered in front of private banks, trying to withdraw their life savings.
Gunfire erupted at one point, though the Afghan presidency sought to downplayed the shooting.
"The defense and security forces along with the international forces working for the security of Kabul city and the situation is under control," the presidency said amid the chaos.
Militants posted photos online early Sunday showing them in the governor's office in Jalalabad, the capital of Nangarhar province.
Abrarullah Murad, a lawmaker from the province told The Associated Press that the insurgents seized Jalalabad after elders negotiated the fall of the government there.
Murad said there was no fighting as the city surrendered.
The militants took also Maidan Shar, the capital of Maidan Wardak, on Sunday, only some 90km from Kabul, Afghan lawmaker Hamida Akbari and the Taliban said.
Another provincial capital in Khost fell later Sunday to the Taliban, said a provincial council member who spoke on condition of anonymity for fear of reprisals.
The fall on Saturday of Mazar-e-Sharif, the country's fourth largest city, which Afghan forces and two powerful former warlords had pledged to defend, handed the insurgents control over all of northern Afghanistan.
Atta Mohammad Noor and Abdul Rashid Dostum, two of the warlords Ghani tried to rally to his side days earlier, fled over the border into Uzbekistan on Saturday, said officials close to Dostum.
They spoke on condition of anonymity as they weren't authorised to publicly speak about his movements.
Writing on Twitter, Noor alleged a "conspiracy" aided the fall of the north to the Taliban, without elaborating.
"Despite our firm resistance, sadly, all the government and the Afghan security forces equipment were handed over to the Taliban as a result of a big organised and cowardly plot," Noor wrote.
"They had orchestrated the plot to trap Marshal Dostum and myself too, but they didn't succeed."
In his speech on Saturday, Ghani vowed not to give up the "achievements" of the 20 years since the US-led invasion toppled the Taliban after the 9/11 attacks.
The US has continued holding peace talks between the government and the Taliban in Qatar this week, and the international community has warned that a Taliban government brought about by force would be shunned.
But the insurgents appear to have little interest in making concessions as they rack up victories on the battlefield.
"We have started consultations, inside the government with elders and political leaders, representatives of different levels of the community as well as our international allies," Ghani said.
"Soon the results will be shared with you," he added, without elaborating further.
Many Afghans fear a return to the Taliban's oppressive rule. The group had previously governed Afghanistan under a harsh version of Islamic law in which women were forbidden to work or attend school, and could not leave their homes without a male relative accompanying them.
Salima Mazari, one of the few female district governors in the country, expressed fears about a Taliban takeover on Saturday in an interview from Mazar-e-Sharif, before it fell.
"There will be no place for women," said Mazari, who governs a district of 36,000 people near the northern city. "In the provinces controlled by the Taliban, no women exist there anymore, not even in the cities. They are all imprisoned in their homes."
In a statement late Saturday, however, the Taliban insisted their fighters wouldn't enter people's homes or interfere with businesses. They also said they'd offer an "amnesty" to those who worked with the Afghan government or foreign forces.
"The Islamic Emirate once again assures all its citizens that it will, as always, protect their life, property and honour and create a peaceful and secure environment for its beloved nation," the militants said.
"In this regard, no one should worry about their life."
Despite the pledge, those who can afford a ticket have been flocking to Kabul International Airport, the only way out of the country as the Taliban took the last border crossing still held by the government Sunday at Torkham.
Pakistan's Interior Minister Sheikh Rashid Ahmed told local broadcaster Geo TV that Pakistan halted cross-border traffic there after the militants seized it.