Nine people had been confirmed dead, a municipal authority official said, while rescuers estimated more than 16 had been injured and taken to hospital.
Rescue workers, police and residents helped pull 13 people out of the rubble and were looking for those buried beneath. The building is located in a congested lane of the Bhendi Bazaar area in southern Mumbai.
Thousands of Mumbai buildings that are more than 100 years old are at risk of collapse, their foundations weakened partly by some of the heaviest rainfall that the city has witnessed in more than 15 years.
Authorities were advising people living in an adjacent building to vacate after it developed cracks following Thursday's collapse.
Building collapses are common in India during the monsoon season, which is June to September. High demand and lax regulations encourage some builders to use substandard materials or add unauthorized extra floors.
Meanwhile, the city was slowly limping back to normalcy after it was paralyzed by heavy downpours for two days.
Train services and public transport were halted and airports shut on Tuesday as roads turned into rivers and floodwaters seeped into many low-lying buildings. In many places, people had to abandon their vehicles and wade through waist-deep water to reach their homes.
Schools, colleges and offices that were shut Wednesday opened Thursday, but attendance was sparse.
Mumbai has been hit by several deadly building collapses in recent years, often caused by shoddy construction, poor quality materials or ageing buildings.
The city is particularly vulnerable to deadly collapse with millions forced to live in cramped, ramshackle properties because of rising real estate prices and a lack of housing for the poor.
Activists say housing societies, private owners and builders often cut corners to save on costs. They also claim that corruption plays a part with officials sometimes knowingly certifying dilapidated buildings in return for money.