The two teenage girls dressed in full hijabs entered the busy market and detonated their explosives, said Abba Aji Kalli, the Borno State coordinator of the Civilian Joint Task Force.
The first set off her explosives and killed about three women, said Kalli.
When others gathered around the scene, the second bomber screamed and blew herself up, killing about 30, he said.
The figures are believed to have risen.
“I am right here at the scene and I have before me 11 corpses … many have been taken away by relatives, while others are taken to the state specialists’ hospital,” said Kalli.
Soldiers and police officers cordoned off the area while rescue workers helped survivors to the hospital. Nigeria’s police have not yet issued a statement on Tuesday’s blasts.
Nigeria’s Islamic militant rebel group, Boko Haram, is suspected of the suicide blasts, as they have carried out many similar violent acts in northeastern Nigeria. More than 1,500 have died so far this year in the extremists’ insurgency, which has been going on for five years, according to Amnesty International.
On Monday, suspected insurgents took over Damasak town, less than 200km (125 miles) from Maiduguri, setting fire to houses and killing an unspecified number of people.
The assailants entered the town disguised as traders, carrying boxes which were packed with AK-47 rifles.
The insurgency has intensified despite the government declaring a state of emergency last year in Borno and neighbouring Adamawa and Yobe states.
Maiduguri is the provincial capital and largest city in Borno.