Abdul Hassib Seddiqi told the BBC's Afghan Service that Mullah Omar had died of health problems at a hospital in Pakistan.
Afghanistan's government says information on his death is "credible".
The latest reports of Mullah Omar's death are being taken more seriously than previous such reports. The Taliban are expected to issue a statement soon.
Sources at the Taliban's two main councils in Quetta and Peshawar in Pakistan told the BBC they were in intensive talks to agree on a replacement for Mullah Omar.
A statement from the office of Afghanistan's President Ashraf Ghani said that it believed, "based on credible information", that Mullah Omar died in April 2013 in Pakistan.
The Afghan government, elected last year, has embarked on a peace process with the Taliban. In its statement, the government called on "all armed opposition groups to seize the opportunity and join the peace process".
A security official in Pakistan, the country hosting the talks, told AP news agency that the claims of Mullah Omar's death were mere "speculation", designed to destabilise the negotiations.
Pakistan has always denied that Mullah Omar was in the country.
The White House says it believes reports of his death are credible.
Who is Mullah Omar?
Lyse Doucet: What is the future for the Taliban?
The Taliban leaders and members of their religious council (shura) have been locked in talks since Tuesday to elect the new supreme leader.
Who they choose is crucial. The selection of his successor will have a big impact on the war and peace in Afghanistan as well as on the future of the Taliban movement itself. The decision will affect the peace talks between the Afghan government and the Taliban.
Mullah Omar's death and the choice of his successor will also have an impact on the unity and cohesiveness of the Taliban.
Mullah Omar was the glue that held the movement together since it was launched in 1994. He had become a mythical figure within the group and was "religiously" obeyed by the ranks and files all along.
The selection of a weak person or someone with a questionable legitimacy could result in the fragmentation of the Taliban and possible defections to the Islamic State.
Mullah Omar led the Taliban to victory over rival Afghan militias in the civil war that followed the withdrawal of Soviet troops.
His alliance with al-Qaeda leader Osama Bin Laden prompted the US-led invasion of Afghanistan in 2001, in the aftermath of the 9/11 attacks on New York and Washington.
Mullah Omar has since been in hiding, with a $10m (Â£6.4m) US state department bounty on his head.
Over the years, the Taliban have released several messages purported to be from the fugitive leader.
The latest of these statements, from mid-July, expressed support for the peace talks between the Taliban and the Afghan government.
However, the message was in the form of a text published on a Taliban website, rather than an audio or video recording - fuelling rumours that the leader was dead or incapacitated.
The failure to prove that Mullah Omar was alive was a major factor behind the defection of several senior Taliban commanders to the so-called Islamic State group, according to the BBC's former Kabul correspondent, David Loyn.
Taliban leader Mullah Omar died two years ago in Pakistan, a spokesman for Afghanistan's security services says.