Military vehicles entered the large compound in Khartoum housing the defence ministry, the army headquarters and Mr Bashir's personal residence to topple him.
British Foreign Secretary Jeremy Hunt says two years of potential military rule in Sudan "is not the answer" for "real change" in the country.
While addressing a news conference in the capital, Khartoum on Firday, Omar Zein Abideen said that the army has "no ambition to hold the reins of power", and stressed that "we are ready to step down as early as a month if a government is formed".
Protesters want a civilian council to lead the transition rather than a military one.
Lt-Gen Omar Zain al-Abidin, who heads the military council's political committee, said on Friday: "The solutions will be devised by those in protest". "The Sudanese people have been clear that they have been demanding a civilian-led transition". He said al-Bashir's top government members, including the vice president and associates, are also under arrest but didn't elaborate.
Ibn Ouf, who is on a US sanctions list for genocide in Darfur, was sworn in as head of the new military transitional council, which also includes Zein Abedeen.
The Sudanese Professionals Association, which has spearheaded the four months of demonstrations against al-Bashir, denounced the military's statements as a "farce".
The 75-year-old, now under house arrest, becomes the second regional leader after Algeria's military-backed president, Abdelaziz Bouteflika, to leave this month in the face of nationwide protests.
Chants by the protesters since the military council's announcement have often focused on how close its members were to Bashir in the very recent past. Organizers said they would keep up the campaign and that they disagree with the army's plans to rule the country for the next two years.
It also vowed to "resist" by peaceful means all the extraordinary measures the military has imposed since Thursday's ouster of al-Bashir, including the nighttime curfew and state of emergency.
Asked whether the United States supported Bashir being put on trial before the International Criminal Court for Darfur atrocities, Palladino said: "We believe that the victims of Darfur deserve justice, that accountability is essential for achieving lasting peace in Darfur".
He pledged the military would stay in power only as long as it was needed. Mohammed Hamadati, the commander, said talks are needed so Sudan would "avoid slipping into chaos".
That came after the African Union decried Bashir's military overthrow, saying it was "not the appropriate response to the challenges facing Sudan and the aspirations of its people".
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