Citing Afghan government and intelligence sources, the BBC said that the Omar died two to three years ago. No further details were released.
Quoting a Taliban spokesman, it said the militant group would issue a statement shortly.
Early this month, the Taliban had issued Omar's message on the eve of Eid in which he had hailed as "legitimate" the July 7 peace talks between his group and the Afghan government aimed at ending 13-year war in Afghanistan.
The Taliban leader is hiding following the ouster of his government in 2001. It was Omar's backing for al-Qaeda chief Osama Bin Laden after the 9/11 attacks that sparked the US-led campaign in Afghanistan.
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He is carrying a USD 10 million bounty on his head.
Taliban leaders and fighters have not heard from their chief since 2007.
Afghan Taliban this year published a surprise 5,000-word biography of Omar to mark his 19th year as the group's chief.
It lists Omar's military feats fighting the Russians between 1983 and 1991, saying he was wounded four times and lost his right eye.
In 1994, he took over leading the militants to tackle the "factional fighting" among warlords that followed the collapse of the communist regime in 1992.
In 1996, Omar was conferred the title "ameer-ul-momineen" (head of the pious believers), becoming Taliban's supreme leader.