Part of the building, located in a crowded suburb in the central part of the city, crumbled around midnight on Monday, the city's civic body BMC said.
Rescue workers were working to pull out debris and rubble until late on Tuesday. Heavy rain hampered the work.
Building collapses are common in Mumbai during monsoon season, mostly due to poor construction.
More than a dozen people were injured in the accident, of which 10 were discharged from hospitals after treatment, the BMC said.
"People living in the building were asked to vacate the houses five to six years ago. Every year a notice was sent to them, but no one left," said Suvrana Bansode, a resident living close to the building that collapsed.