Three Taliban sources said the Islamist militia had on Friday seized the Panjshir valley north of Kabul, the last part of Afghanistan holding out against it.
"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," said one Taliban commander.
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It was not immediately possible to confirm the reports.
Former vice president Amrullah Saleh, one of the leaders of the opposition forces, told the television station Tolo News that reports that he had fled the country were lies.
And in a video clip posted on Twitter by a BBC World journalist who said it had been sent by Saleh, he said: "There is no doubt we are in a difficult situation. We are under invasion by the Taliban ... We have held the ground, we have resisted."
He also tweeted: "The RESISTANCE is continuing and will continue. I am here with my soil, for my soil & defending its dignity."
His son, Ebadullah Saleh, denied that Panjshir had fallen, texting the message "No, it's false".
There had been reports of heavy fighting and casualties in Panjshir, a rugged valley where several thousand fighters from regional militias and remnants of the old government's armed forces had massed under the leadership of Ahmad Massoud, the son of late Mujahideen commander Ahmad Shah Massoud.
The Taliban seized Kabul on Aug. 15 after rapid advances across Afghanistan.