A civilian militia source in Maiduguri, capital of Borno state, told AFP "at least 13" blasts were heard throughout the city at dawn, some two hours before voting opened at 0700 GMT.
Residents also reported hearing the explosions.
"Boko Haram terrorists attempted to enter the city but were intercepted," a security source told AFP.
"They resorted to firing RPGs (rocket-propelled grenades) on the city. Attempts have been made to locate the source of the firing."
A second source said one of the explosions occurred at the headquarters of the Nigerian Army 7th Division as troops were preparing to go out on operations.
"One soldier was killed and 20 were injured. There were other explosions at Maimalari barracks but there was no casualty," he added.
A fighter jet was seen flying over the city, and the source said troops were searching for the source of the fire.
Three mortar shells fell into an open rice field in the Teachers Village camp for people displaced by the conflict but there were no immediate reports of casualties.
But the army denied there had been any attack, saying the city was "calm and peaceful".
Borno state police commissioner Damien Chukwu acknowledged the sound of gunfire but said it was "for security purposes", insisting there was "no threat to public peace and order".
There have been growing fears of a major Boko Haram attack in the run-up to the election, given the jihadists' previous pledges to disrupt the poll.
Maiduguri is the birthplace of Boko Haram and has been repeatedly attacked during its nearly 10-year insurgency that has devastated the remote region.
Wave of attacks
Voters were on Saturday casting their ballots in Nigeria's presidential election, with incumbent Muhammadu Buhari, a former army general who has vowed to defeat the jihadists, seeking a second term in office.
At about the same time as they attacked Maiduguri, the jihadists also moved on Geidam, north of the Yobe state capital Damaturu, but were repelled by troops who were alerted by locals, security sources said.
There were also clashes in Auno and Goniri, on both sides of the border between Borno and Yobe.
Elsewhere in Borno, which has been worst hit by the conflict, Boko Haram fired rockets at the town of Gwoza, injuring three people and partially damaging a mosque.
There was also an attack late on Friday on Zabarmari village, 10 kilometres (six miles) outside Maiduguri.
In the Borno town of Gamboru, on the border with Cameroon, Chadian troops in about 50 vehicles, including tanks, arrived at about 11:30 am on Saturday, while voting was in progress.
The Chadians are part of the regional force fighting Boko Haram. N'Djamena said on Friday night it had deployed more than 500 soldiers to support their Nigerian counterparts.
Boko Haram's insurgency in northeast Nigeria has killed more than 27,000 people and left 1.8 million homeless, creating a humanitarian crisis.
But the government and military have repeatedly said the jihadists are weakened to the point of defeat, despite persistent hit-and-run raids and suicide bomb attacks.
Recent months have seen an increase in attacks against troops, blamed on or claimed by Boko Haram's Islamic State West Africa Province (ISWAP) faction.
The IS-backed group is active in northern Yobe state and around the border with Borno while the faction led by Abubakar Shekau mostly operates east and south of Maiduguri.