Mullah Akhtar Mohammad Mansour was appointed leader at a meeting of the Taliban's top representatives, according to the sources who were present at the shura, or gathering, Reuters reported.
"The shura held outside Quetta (Pakistan) unanimously elected Mullah Mansour as the new emir of the Taliban," said one commander at the Wednesday night meeting.
Siraj Haqqani, leader of the powerful Haqqani militant faction, will be a deputy to Mansour, according to two Taliban commanders who spoke to Reuters.
Mansour will be only the second leader the Taliban has had, following Omar, who founded the ultra-conservative Islamist movement in the 1990s.
On Thursday, Taliban spokesman Zabihullah Mujahid confirmed that Omar had died. His statement, however, indicated that Omar died recently, rather than more than two years ago as the Afghan government said on Wednesday.
"For some time, (Omar) has been suffering a kind of sickness and over the last two weeks it became more serious, and due to that illness he passed away," Mujahid said.
Afghanistan had said Omar died in April, 2013 in a Pakistani hospital, but Pakistani officials could not confirm that. Omar had not been seen since 2001, and the Taliban said he had never left Afghanistan in the last 14 years.
The announcement of Omar's death threw into disarray the peace process being fostered by Pakistan aimed at ending nearly 14 years of violence between the Taliban and the Western-backed Afghan government.
Mujahid said earlier on Thursday that the Taliban's official team of negotiators was "not aware of this process" in Pakistan, Reuters reported.
Later in the day, the Pakistani foreign office said a planned second round of meetings set for Friday would be delayed at the request of the Taliban leadership. The two sides held inaugural talks in Pakistan earlier this month.
"Pakistan and other friendly countries of Afghanistan hope that the Taliban leadership will stay engaged in the process of peace talks in order to promote a lasting peace in Afghanistan," the foreign ministry statement said.