Under new Taliban chief Mullah Akhar Mansoor, Omar's son Yaqoob and brother Abdul Manan were offered memberships of the military commission and the powerful leadership council respectively, the Express Tribune reported.
However, the two leaders are upset over assaults against renegade Taliban militants who refused to accept Mansoor as leader and set up their own group under leadership of Mullah Muhammad Rasool this month, the paper added.
The Taliban last week launched a major operation in Zabul province of Afghanistan, killing several members of the new breakaway faction.
Mullah Omar's family is not happy with the killing of the Taliban militants in infighting and may refuse to be part of the group led by Mansoor, the paper said.
The family had initially opposed Mansoor's election but later pledged allegiance to him, apparently putting at rest fears of a split within the group.
Citing sources, the paper said Mansoor's commanders are planning another offensive against Rasool and his supporters in Farah and Herat provinces, where they are currently based.
It said Taliban's Helmand governor Haji Abdul Manan, also known as Haji Abdul Rahim Akhund, may lead the operation.
Taliban spokesman Zabihullah Mujahid denied the reports of operation, saying "There is no such plan."
The development came at a crucial time as refusal of Omar's close relative to work with Mansoor will deprive him of legitimacy when he is being pressurised by Pakistan to open peace talks with the Afghanistan government.
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