One of the resistance leaders, Amrullah Saleh, told the BBC the situation was “difficult”.
In a video message, Mr Saleh, a former vice-president of Afghanistan, said: “There is no doubt we are in a difficult situation.
“We are under invasion by the Taliban.”
But he added: “We will not surrender, we are standing for Afghanistan.”
News of the heavy fighting comes as Taliban sources claimed they had seized the area from the rebels.
However, resistance fighters have denied the claims.
One Taliban commander told Reuters: “By the grace of Allah Almighty, we are in control of the entire Afghanistan.
“The troublemakers have been defeated and Panjshir is now under our command.”
Having taken over the remnants of the US equipment left over from their chaotic withdrawal from the country, the Taliban now commands a huge arsenal of weaponry and machinery, including helicopters, tanks and munitions.
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On Wednesday, Ahmad Massoud, leader of the resistance fighters revealed the Taliban had offered the rebels “one or two seats” in their new administration, but he had rejected the deal.
Mr Massoud is the son of the late guerilla commander Ahmad Shah Massoud who held out against the Soviet invasion of Afghanistan in the late 80s and the Taliban in the 90s.
Senior Taliban official Amir Khan Muttaqi issued an audio message to the rebels on Wednesday claiming Taliban forces had surrounded the valley.
He also called on the local people of the Panjshir to tell fighters to lay down their arms.
“Those who want to fight, tell them it is enough,” Mr Muttaqi said.