Explosions and gunfire could be heard after the fighters overwhelmed a checkpoint at the edge of the city.
The government has said the crackdown is part of a campaign to ensure that foreigners do not participate in the closely fought vote.
The Nigerian military said it couldn't guarantee an election commissioner's safety unless the vote was held back for six weeks.
The insurgency in the northeast is causing chaos ahead of the postponed election, but so is the oil price plunge.
The education activist urged Nigeria's incoming government to make rescuing the girls a priority.
The president and legislative elections will be pushed back six weeks to give a multinational force time to fight Boko Haram militants.
The country’s military is the primary backer of a postponement, arguing that it would allow them more time to fight Boko Haram.
Ahead of Nigeria's elections, Boko Haram is stepping up its campaign of violence — and using women to carry it out.
Violence has escalated in Nigeria in the weeks ahead of the country's presidential election Feb. 14.
Demonstrators have demanded that the country's upcoming election be postponed to avoid disenfranchising millions of Nigerians.
No one has yet taken responsibility for the lethal bomb in the Nigerian city of Gombe, though Boko Haram has already struck there before.
The assault is the second the group has launched against the strategically important city of Maiduguri in the past week.
The potential disenfranchisement of up to 1.5 million people could undermine the credibility of the already divisive election.
The extremist group is reportedly engaged in fierce fighting with the Nigerian military, which is responding with airstrikes and ground troops.
Secretary of State John Kerry is expected to speak with Nigerian President Goodluck Jonathan and leading challenger Muhammadu Buharim.
Technical issues struck about 10 million people from voter rolls, leaving many with no voter ID cards ahead of the ballot.