The fire started on Saturday night at the BM Container Depot in Sitakunda, in Chittagong District, the Bangladesh Sangbad Sangstha (BSS) agency reported.
Police and fire service personnel are trying to douse the flames and rescue people.
The death toll could rise as some of the injured are in critical condition, said Chittagong civil surgeon Mohammed Elias Hossain, according to the Reuters news agency. He added that firefighters and police officers were among those injured.
Divisional commissioner of Chattogram, Asharf Uddin, said that families of those killed would be compensated with 50,000 Bangladeshi taka ($560), while those injured and their families will be compensated as well.
The incident is the latest in a string of fatal industrial accidents over several years in the South Asian nation.