In the first blast, 25 people were killed while the second one, according to Police Commissioner Shehu Umar, claimed more than 14 people, Xinhua reported.
The police chief added that one man dressed as a woman has been arrested in connection with the blasts.
A security source said that most of the people who died in the first blast were adherents of the Darika Islamic sect led by Sheikh Dahiru Bauchi, a renowned Islamic cleric.
The first explosion was at the Alkali Kaita Road while the second one occurred near the Kawo bridge, close to an army base.
Former head of state and chieftain of the All Progressive Congress party, Maj. Gen. Muhammadu Buhari, was said to have narrowly escaped the second blast.
Most of the corpses were conveyed to the Nigerian Army Reference Hospital in the city and other hospital in the state.
The state government has declared a 24-hour curfew in the city, to forestall any breakdown of law and order.
The government described the blasts as acts of cowardice and pledged that the government would uncover the perpetrators.