The Syrian Observatory for Human Rights said the attack hit the Muwasalat district of Shaar neighbourhood in the rebel-held east of the city.
The group said the toll was expected to rise, and that the second barrel bomb had been dropped at the scene shortly after the first one hit.
A video of the aftermath of the strikes, posted by the Observatory online, showed ambulances arriving and emergency workers carrying stretchers rushing into a street shrouded in thick white smoke.
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Women could be heard screaming hysterically at the scene of the attack.
Aleppo has been divided between rebel control in the east and regime control in the west since shortly after fighting began there in mid-2012.
Syria's military has increasingly resorted to using so-called barrel bombs, which rights groups have condemned as a particularly indiscriminate weapon that often kills civilians.
Barrel bombs are typically constructed from large oil drums, gas cylinders or water tanks filled with high explosives and scrap metal.
According to the Observatory and Syrian activists, the regime has stepped up its use of the crude weapons in recent weeks, as international attention focuses on the fight against the Islamic State jihadist group.