Taliban and opposition forces are fighting for control of the Panjshir valley north of Kabul, the last province in Afghanistan holding out against the Islamist militia.
Taliban sources had said on Friday the group had seized control of the valley, although the resistance denied it had fallen.
The Taliban has so far issued no public declaration that it had taken the valley, which resisted their rule when it was last in power in Kabul in 1996-2001.
A spokesman for the National Resistance Front of Afghanistan, which groups opposition forces loyal to local leader Ahmad Massoud, said Taliban forces reached the Darband heights on the border between Kapisa province and Panjshir but were pushed back.
"The defence of the stronghold of Afghanistan is unbreakable," Fahim Dashty said in a tweet on Saturday.
A Taliban source said fighting was continuing in Panjshir but the advance had been slowed by landmines placed on the road to the capital Bazarak and the provincial governor's compound. "Demining and offensives are both going on at the same time," the source said.
It was not immediately possible to get independent confirmation of events in Panjshir, which is walled off by mountains except for a narrow entrance and had held out against Soviet occupation as well as the previous Taliban government.
Celebratory gunfire resounded all over Kabul on Friday as reports spread of the Taliban's takeover of Panjshir, and news agencies said at least 17 people were killed and 41 injured in the firing.
Pakistan's spy chief Lieutenant General Faiz Hameed flew into Kabul on Saturday, sources in both capitals said.
It was not clear what his agenda was, but a senior official in Pakistan had said earlier in the week that Hameed, who heads the powerful Inter-Services Intelligence (ISI) agency, could help the Taliban reorganise the Afghan military.
Washington has accused Pakistan and the ISI of backing the Taliban in the group's two-decade fight against the US-backed government in Kabul, although Islamabad has denied the charges. After the Islamist group seized Kabul this month, analysts have said Pakistan's role in Afghanistan will be much enhanced.
Pakistan's government has said that its influence over the movement has waned, particularly since the Taliban grew in confidence once Washington announced the date for the complete withdrawal of US and other foreign troops.
The Taliban source also said the announcement of a new government would be pushed back to the next week.
Earlier, other Taliban sources said the group's co-founder Mullah Abdul Ghani Baradar would lead a new Afghan government set to be announced soon.
Baradar would be joined by Mullah Mohammad Yaqoob, the son of late Taliban co-founder Mullah Omar, and Sher Mohammad Abbas Stanikzai in senior positions, three sources said.
Meanwhile, there were some signs of normality creeping back in the Afghan capital.
Qatar's ambassador to Afghanistan said a technical team was able to reopen Kabul airport to receive aid, according to Qatar's Al Jazeera news channel, which also cited its correspondent as saying domestic flights had restarted.
The airport has been closed since the United States completed operations on Aug. 30 to evacuate diplomats, foreigners and Afghans deemed at risk from the Taliban. However, tens of thousands of people could not be flown out.
The Taliban's main spokesman, Zabihullah Mujahid, also said that one of the main foreign exchange dealers in the capital had reopened.
Impoverished Afghanistan's economy has been thrown into disarray by the takeover by the Taliban. Many banks are closed and cash is in short supply.
The United Nations has said it will convene an international aid conference in Geneva on September 13 to help avert what UN Secretary General Antonio Guterres called a "looming humanitarian catastrophe".
Without the aid that has sustained the country for years, the Taliban will find it hard to avert economic collapse.
Western powers say they are prepared to engage with the Taliban and send humanitarian aid, but that formal recognition of the government and broader economic assistance will depend on action - not just promises - to safeguard human rights.
Australian Associated Press