Abdel Fattah al-Burhan Abdelrahman, the head of the ruling military council, made the comments during a televised address on Saturday.
In the speech he promised the armed forces would consult opposition groups about civilian leadership and lift a curfew imposed by his predecessor.
The lieutenant general also said that a military-led transitional period would last for two years at most.
At least 16 civilians have been killed during the protests but opposition groups have vowed to continue the demonstrations.
The protesters were killed on Thursday and Friday, while a further 20 were shot and left injured, a police spokesperson said.
The coalition of opposition groups leading the demonstrations agreed to meet with the armed forces on Saturday, to discuss a future civilian government.
Sudan Professionals Association (SPA), the group which is leading the demonstrations, had earlier urged protesters to keep marching to demand civilian rule.
The demonstrators have consistently called for rapid political change in Sudan.
“We assert that our revolution is continuing and will not retreat or deviate from its path until we achieve ... Our people’s legitimate demands of handing over power to a civilian government,” the SPA said in a statement.
Protest leaders have already rejected the two-year transition period mentioned by Mr Burhan.
Some are concerned that the military is planning what is known as an “Egyptian scenario”, where an unpopular president is replaced by an equally disliked one from army ranks.
The weeks of protest were initially triggered by a mix of rising unemployment, food costs and government repression.
Mr Bashir was ousted on Thursday and arrested by the military after three decades in power.
General Ahmed Awad Ibn Auf, the defence minister, resigned as head of the transitional military council on Friday after just one day in the post.
He claimed he did so to “preserve unity” in the armed forces and named Mr Burhan as his successor in a televised speech.
Salah Abdallah Mohamed Saleh, Sudan’s intelligence chief, has also resigned, state media reported on Saturday.
Demonstrators had held him responsible for the brutal crackdown on protesters and for the civilian deaths.
Gen Ibn Auf’s resignation prompted jubilation in the streets of Khartoum on Friday.
Thousands of people waved flags and mobile phones and drivers hooted horns through the night.
“Islamists have now lost control and they are in shock. Their ability to project influence in an organised way inside the state appears weak,” Khalid al-Tijani. a Sudanese analyst, said.
“The reason for the changes in Sudan is the pressure from protesters and pressures within the army, and the fear among military commanders of a split in the armed forces.”
Additional reporting by agencies
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