The first explosion targeted moderate Islamic cleric Dahiru Bauchi while the second one targeted senior opposition leader Muhammadu Buhari, a BBC reporter in the city says.
Both are said to have escaped unhurt.
Militant Islamist group Boko Haram has carried out a wave of bombings and assassinations in Nigeria since it launched a brutal insurgency in 2009.
It often targets Muslim leaders opposed to its militant ideology.
There are body parts and damaged cars on the busy Alkali Road in the city centre where the bomb targeting Mr Bauchi exploded, reports the BBC’s Abdullahi Kaura Abubakar from the scene.
It is unclear how many people were killed in the second blast.
Mr Bauchi had completed a preaching session in the nearby Murtala Muhammed square, and was driving through the area in an open-roofed vehicle, greeting well-wishers when he was targeted.
Followers of the renowned cleric have reacted angrily, throwing stones at the security forces and accusing them of failing to protect Nigerians, our reporter says.
The security forces retaliated by firing tear gas.
About 90 minutes after the first attack, a second explosion ripped through the Kawo area, targeting the motorcade of Gen Buhari, a former military ruler of Nigeria and a senior member All Progressive Congress opposition party.
Gunmen rammed a vehicle into his convoy, firing shots at it, our reporter says, adding that two of Gen Buhari’s bodyguards were slightly wounded in the attack.
In May, the emir of the northern area of Gwoza, Shehu Mustapha Idris Timta, was shot dead in an attack blamed on Boko Haram.