The three-story buildings were part of a large complex of similar structures built in 2001 for low-income families in the Madhavnagar area of Vadodara in western Gujarat state, officials said.
Although in different blocks of the complex, the collapse of one building early in the morning was followed minutes later by the other as residents slept.
The Times of India said a search and rescue effort was under way.
The Indian army and the civilian National Disaster Response Force is aiding in the rescue operations, the Press Trust of India reported.
The buildings were planned by the Vadodara Urban Development Authority but residents blamed inferior quality of construction for the disaster, the Times said. One resident said residents had been making complaints to the housing authority but no action had been taken.
"The incident has been taken very seriously and investigations will be conducted," district official Vinod Rao said.
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