Bakri Hassan Saleh, a former interior and defence minister, was named first vice-president as Hassabo Mohammed Abdel Rahman became second vice-president, senior party official Rabbie Abdelatti Ebaid told AFP.
"Yes, confirmed," Ebaid said of the appointments at the start of a government reshuffle.
Saleh was presidential affairs minister in the cabinet which Bashir dismissed last week after the most serious split in years within his ruling National Congress Party (NCP).
Abdel Rahman had been the NCP's political secretary.
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He called Saleh "an efficient and sinister defender of the revolution" who was entrusted with rebuilding the country's intelligence apparatus.
Ebaid said final composition of the new cabinet would be declared within 24 hours, including ministers from other political parties.
Bashir had announced yesterday that his long-time first vice president Ali Osman Taha, a key figure behind the coup, had resigned to pave the way for a new government.
"Ali Osman will voluntarily step down", as he did in 2005 following the signing of a peace agreement that ended 22 years of civil war, Bashir was quoted by the official SUNA news agency as saying.
The president hinted in mid-November that a wide-ranging government shakeup was imminent, after his party moved to expel a group of dissidents led by his ex-adviser Ghazi Salahuddin Atabani.
Critics of Bashir's regime have become increasingly vocal since the government slashed fuel subsidies in September, leading to the worst urban unrest of his rule.
Security forces are believed to have killed more than 200 demonstrators, Amnesty International said, but the government has given a toll of fewer than half that.
Bashir has since talked of "reform", and repeated a call for a dialogue with all political parties, including armed rebels.
Taha led the National Islamic Front party which backed the coup that installed Bashir.
He later became first vice-president but stepped aside for former rebel leader John Garang in July 2005, under the terms of the peace deal which eventually led to South Sudan's independence two years ago.