The fire in the suburb of Vikhroli began around 3 a.m. and was put out 3 hours later, the NDTV channel reported.
The injured were taken to hospitals and other residents were evacuated.
Officials said the fire may have started in an electrical cable box.
The building had housed people moved from the city's slums under a rehabilitation program. .
The CNN-IBN channel quoted some of the residents as saying the building had no emergency exits.