"We have information that one person is dead. Rescue operations are on, but they are difficult due to the heavy rain," a police inspector in the Mahim neighbourhood, where the collapse happened, told AFP.
Four diggers and a crowd of people were gathered around a high mound of rubble -- the remains of half of the five-storey building that caved in, said an AFP photographer at the scene. The other half of the building was still standing.
The Press Trust of India news agency, quoting a municipal officer, said the deceased was an elderly woman and that four others had been injured and taken to hospital.
The accident comes two months after a seven-storey building collapsed on Mumbai's outskirts and killed 74 people, leading to the arrest of two builders and seven others in connection with the tragedy.
The collapse of the partially-finished building highlighted widespread shoddy construction standards in India, where huge demand for housing and pervasive corruption often mean cost-cutting and a lack of safety inspections.
NDTV cited neighbours saying at least 20 people were feared trapped beneath the debris after Monday's collapse.
Downpours on the first two days of the monsoon season in Mumbai have caused flooding, train delays and traffic jams across the city.
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