Districts of Khinj and Unabah have been taken from so-called National Resistance Front, a Taliban spokesperson says, giving Afghanistan's new rulers control of four of province's seven districts.
Fresh fighting has been reported between the Taliban and militia forces in Afghanistan's Panjshir Valley as Taliban claims controlling four of seven districts of the legendary valley.
The Taliban appear determined to snuff out the Panjshir resistance before announcing who will lead the country in the aftermath of Monday's US troop withdrawal, which was supposed to end two decades of war.
But Panjshir, which held out for nearly a decade against the Soviet Union's occupation and also the Taliban's first rule from 1996-2001, is stubbornly holding out.
Fighters from the so-called National Resistance Front (NRF) – made up of anti-Taliban militia and former Afghan security forces – are understood to have stockpiled a significant armoury in the valley, around 80 kilometres north of Kabul and guarded by a narrow gorge.
Taliban spokesman Bilal Karimi said on Saturday the districts of Khinj and Unabah had been taken, giving Taliban forces control of four of the province's seven districts.
"The Mujahideen (Taliban fighters) are advancing toward the centre (of the province)," he said on Twitter.
But the NRF, grouping forces loyal to local leader Ahmad Massoud, said it surrounded "thousands of terrorists" in Khawak pass and the Taliban had abandoned vehicles and equipment in the Dashte Rewak area.
Front spokesman Fahim Dashti added "heavy clashes" were going on.
In a Facebook post, Massoud insisted Panjshir "continues to stand strongly".
Praising "our honourable sisters", he said demonstrations by women in the western city of Herat calling for their rights showed Afghans had not given up demands for justice and "they fear no threats".
Earlier, a Taliban source said the group's advance was slowed by landmines placed on the road to the provincial capital, Bazarak.
Perilous position of NRF
In Panjshir, former vice-president Amrullah Saleh, holed out alongside Massoud – the son of legendary anti-Taliban commander Ahmad Shah Massoud – admitted the perilous position of the NRF.
"The situation is difficult, we have been under invasion," Saleh said in a video message.
Usually known for his sharp Western suits, Saleh was filmed wearing a traditional shalwar kameez tunic and a flat woollen pakol cap favoured by Panjshiris.
"The resistance is continuing and will continue," he added.
Taliban and militia tweets suggested the key district of Paryan had changed hands several times in the last few days, but that also could not be independently verified.
Away from the valley, the international community was coming to terms with having to deal with the new Taliban government with a flurry of diplomacy.
US Secretary of State Antony Blinken is due on Sunday in Qatar, a key player in the Afghan saga and the location of the Taliban's political office, though he is not expected to meet with the group.
He will then travel to Germany, to lead a virtual 20-nation ministerial meeting on Afghanistan alongside German Foreign Minister Heiko Maas.
Pakistan's intelligence chief Faiz Hameed was in Kabul, meanwhile. Hameed was reportedly in the city to be briefed by his country's ambassador but is also likely to meet top Taliban officials with whom Islamabad has historically had very close relations.
UN Secretary-General Antonio Guterres is also set to convene a high-level meeting on Afghanistan in Geneva on September 13, to focus on humanitarian assistance for the country.
The United Nations has already restarted humanitarian flights to parts of Afghanistan, while the country's flag carrier Ariana Afghan Airlines resumed domestic trips on Friday and the United Arab Emirates sent a plane carrying "urgent medical and food aid".
Western Union and Moneygram, meanwhile, said they were restarting cash transfers, which many Afghans rely on from relatives abroad to survive.
China has already confirmed it will keep its embassy in Kabul open.