Torrential downpours have swept away homes, triggered landslides and claimed at least 180 lives across South Asia.
"Two people have been declared dead and over 40 are trapped under the debris," disaster management spokesman Tanaji Kamble told AFP of the building collapse in southern Mumbai's congested Dongri area.
Kamble said five others, including two children, were rescued and taken to hospital.
Earlier this month, a wall collapsed in the city, killing 29 people.
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Two teams from India's National Disaster Response Force (NDRF) as well as local volunteers, fire department and police officials were scouring the rubble for survivors.
"The building collapse sounded like an earthquake and we rushed to the spot to check the loud noise," a local resident told Indian news channel NDTV.
Another resident said the building was more than a century old and had 10 families living inside despite its dilapidated condition.
Building collapses in Mumbai, home to 20 million people, are common during the monsoon season because of rickety structures that buckle under the weight of continuous rain.
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