The blasts in Maiduguri and dawn raid in Damaturu came just days after a deadly mosque attack in the northern city of Kano left at least 120 people dead and hundreds more injured.
Almost daily attacks have heightened tensions in Nigeria's far northeast and sparked fears of unrest beyond the country's borders.
Both Maiduguri, which is the capital of Borno state, and Damaturu, the main city in neighbouring Yobe, have been hit repeatedly by Boko Haram.
In that attack, more than 45 people were killed.
Two suspected female suicide bombers were arrested in the city on Wednesday and Thursday, while on Friday, a roadside bomb near another market was defused.
Boko Haram claimed responsibility for a similar raid in Damaturu on October 24, 2013, in which four police buildings were hit with guns and explosives and militants battled with security forces.
Both incidents will add to claims that Nigeria's government has not done enough to protect civilians from the Islamists, whose insurgency has claimed over 13,000 lives since it began in 2009.
In Maiduguri, witnesses said the first explosion happened as a middle-aged woman tried to enter the market and refused a security check of her bags.
A second woman then tried to get into a nearby shop but as she was denied access detonated her explosives.
Death tolls are notoriously difficult to corroborate in Nigeria and the authorities often downplay numbers.
Borno state police commissioner Clement Adoda said six people were killed, including the two bombers, but one witness said he counted at least 10 dead bodies at the scene.
Borno, Yobe and neighbouring Adamawa were placed under a state of emergency in May 2013 but the special measures have failed to end the violence.
Boko Haram has seized control of more than two dozen towns in all three states and more than 1.5 million Nigerians have been forced to flee their homes, the United Nations said last week.
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