Most of the victims were asleep when the fire started about 3 a.m. in a six-story building, the Hindustan Times reported.
Mamata Banerjee, chief minister of West Bengal State, of which Kolkata (formerly called Calcutta) is the capital, said the fire was caused by "unplanned construction" and that the building had no proper fire exits.
Wires intertwined on the ground floor caused a short circuit that sparked the fire, a preliminary investigation indicated. The fire spread to other parts of the building.
Some 26 fire engines worked to contain the fire.
Residents, however, said firefighters took 40 minutes to arrive.
Fire service minister Javed Khan responded that construction of the market complex had been unauthorized and didn't follow fire safety standards. He said action would be taken against the market authorities.
NDTV television said the Surya Sen Market complex housed several shops and plastic and paper warehouses. The report said most of those killed were owners of the facilities.
The report said rescue operations were hampered by heavy smoke in the narrow lanes leading to the building.