The force's leader Mohammed Hamdan Dagolo, known as Hemedti, reportedly personally resents Awad Ahmed ibn Auf, the new leader of the military council who announced Bashir's arrest on Thursday.
The state news agency SUNA said ibn Ouf planned to meet with political factions and leaders of the protest movement later Friday and that he was holding off with announcing the full makeup of the military council pending the consultations.
The U.S. placed him under sanctions in 2007 for his role in the conflict in Sudan's western Darfur region, where he liaised between the government and the Janjaweed, a militia notorious for its attacks on civilians.
Al-Bashir is wanted by the International Criminal Court for alleged war crimes and genocide in Darfur.
His downfall followed months of unrest which began in December over the rising cost of living.
United Nations chief Antonio Guterres called for a transition that would meet the "democratic aspirations" of the Sudanese people and appealed for "calm and utmost restraint by all", his spokesman said.
Ibn Auf also announced a state of emergency, a nationwide ceasefire and the suspension of the constitution.
Youssef pointed out that there was no legal basis for forming a military council and called on all political forces, and the army to reach agreement on the formation of a government. Bashir acknowledged that the demands of protesters demonstrating against his government were "legitimate" but were expressed unlawfully causing several deaths.
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Demonstrators who have been holding nearly daily anti-Bashir protests have rejected the decision to set up a transitional military council and vowed to continue protests until a civilian government is established.
The announcement on Thursday of military control for a two-year "transitional period" was immediately rejected by protesters who have for nearly four months been demanding Bashir be removed from office and are now calling for the military to hand over power to civilians. "We are not greedy for power".
In a challenge to the military council, several thousand protesters remained in front of the defense ministry compound, and in other parts of the capital, as the curfew went into effect.
"So what we need to do is to continue the fight and the peaceful resistance", she said.
"First of all we don't know who's behind the military council", she said.
Since returning to Khartoum from self-imposed exile, Mahdi has allied his party with the grass-roots momentum driving the mass protests that preceded Thursday's coup. He said Mr Bashir was being held "in a secure place" but did not give details.
He promised people that the army will hand over to a democratic regime after a two-year transitional period.
UK Foreign Secretary Jeremy Hunt said that a two-year military council was "not the answer" and called for a "swift move to an inclusive, representative, civilian leadership".