The protests were initially fueled by anger over the deteriorating economy but quickly turned to demands for the president's ouster, and gained momentum last week after Bouteflika's resignation in Algeria.
Security forces have responded to the protest movement with a fierce crackdown, killing dozens.
In a statement, the Sudanese defence minster and army general Ahmed Awad Ibn Auf said Bashir, who has ruled for 30 years, had been arrested "in a safe place". Presidential elections will be held after the transition period expires.
Government sources said Bashir had stepped down and consultations were underway to form a transitional council.
Mr Al-Bashir was pushed out of office following five days of massive protests across the country.
It urged people to continue the sit-in outside military headquarters and to stay on the streets of cities across the country. He called on all armed groups to join the government and protect the people.
Earlier in the day, tens of thousands of protesters celebrated jubilantly in the streets of Khartoum, shouting "he's gone, he's gone", after widespread reports that Mr.al-Bashir had been arrested overnight after the months of demonstrations.
British Foreign Secretary Jeremy Hunt called for a "swift move to an inclusive, representative, civilian leadership", saying in a tweet that a "military council ruling for two years is not the answer".
"Ocampo suggested if Bashir's stash of money were disclosed (he put the figure at possibly $9 billion), it would change Sudanese public opinion from him being a "crusader" to that of a thief", he said, referring to International Criminal Court Prosecutor Luis Moreno-Ocampo.
At first the emergency rule curbed the scale and intensity of the protests, but before long demonstrators staged a massive rally outside the military headquarters that reverberated with chants of "overthrow, overthrow".
The transitional council will be headed by Auf, Al-Arabiya TV earlier reported.
The statement called on the worldwide community to help Sudan to have a peaceful transition.
Sudan's state security agency announced the release of all political prisoners across the country, but did not say when.
Police have ordered officers not to intervene against the protests.
Troops deployed around the defence ministry and on major roads and bridges in the capital.
Bashir was backed by the National Islamic Front of his then mentor, the late Hassan al-Turabi. They were reportedly shouting "It has fallen, we won", Reuters said.
The protests that erupted in December have been the biggest challenge to his rule.
"We won't leave from here until we know what it is".
One senior USA official appeared to indicate that if the extent of Bashir's embezzlement were known, it could destroy much of the public's trust in his administration.
Mr Ibn Auf has also been accused of complicity in genocide, as a 2007 US Treasury sanctions notice detailed his "role in fomenting violence and human rights abuses in Darfur".
Sudanese protesters take part in an anti-government demonstration in Khartoum, Feb. 14, 2019.
There has been an often festive mood at the sit-in, with protesters singing and dancing to the tunes of revolutionary songs.