The victims were migrant workers from former Soviet states, AFP news agency quoted the city's emergencies ministry as saying.
The two storey building was being used as living quarters for the market vendors.
The fire at Kachalovsky market came hours after a blaze at a skyscraper in Moscow, in which no-one was hurt.
The market fire broke out at about 04:50 (00:50 GMT) and was extinguished some three hours later, RIA Novosti said, citing officials.
There are unconfirmed reports that the people who died came from the central Asian nation of Tajikistan.
Emergency workers described squalid, cramped living conditions where people slept on hard cots stacked on top of each other, according to the Interfax news agency. Read More