The Taliban said they had taken some territory and inflicted “heavy” losses on the National Resistance Front (NRF).
But the NRF said it had control of all entrances to the valley, and the Taliban had lost hundreds of fighters.
Panjshir is the only province that has not fallen to the Islamist group and thousands of opposition fighters are believed to have massed there.
The resistance – which includes former Vice-President Amrullah Saleh, former Afghan security force members and local militias – is led by local tribal leader Ahmad Massoud. Massoud’s father held off the Soviets in the 1980s and the Taliban in the 1990s.
Panjshir – the valley trying to hold off the Taliban
Meanwhile US President Joe Biden visited troops who were injured in a devastating bomb attack at Kabul airport as they helped to evacuate Americans and Afghans from the country. Thirteen US soldiers were among some 170 people killed.