Nobody has claimed responsibility for the blasts, which went off around 5:30 p.m. local time (1630 GMT), but suspicion is likely to fall on the militant Islamist group, which has attacked the city several times during a six-year insurgency that has killed thousands.
A Red Cross official, who was involved in the evacuation, and a National Emergency Management Agency representative said about 50 people were killed. Both asked to remain anonymous.
Around 70 people were injured in the attacks, the Red Cross official added.
President Muhammadu Buhari has made crushing Boko Haram his top priority, but hundreds have been killed in bombings and attacks concentrated in the northeast of Africa's top oil producer since he was inaugurated on May 29.
Buhari replaced his defense chiefs on July 13 as part of an attempt to step up the campaign against Boko Haram, which is trying to establish a state adhering to strict Sharia law in the country's predominantly-Muslim northeast.
The radical Islamist group controlled a swathe of land around the size of Belgium at the end of 2014. Nigeria earlier this year managed to push the militants out of most of that territory with the help of troops from neighboring Chad, Niger and Cameroon.
But the insurgents were dispersed and in recent weeks have carried out a string of attacks on targets in northern Nigeria as well as the countries with which it shares borders.
Nigeria's president has worked with regional counterparts to set up a multinational joint task force involving troops from the neighboring countries affected by the insurgency.