Official said 11 people trapped under the debris were rescued and one of them died. The other injured were admitted to the nearby the Ramachandra Medical College.
The incident occurred at Moulivakkam near Porur area, about 20 km from Chennai.
12 fire tenders and the National Disaster Management Authority team reached the site. Senior police and other officials were also on the spot.
Police took two people in custody in connection with the tragedy.
Most of the people who were trapped under the debris are said to be construction workers from Andhra Pradesh while some might have taken shelter as it was raining at the time of the collapse,, local witnesses said.
Rescue efforts were hampered early due to power cut and rains. However, officials later ensured that the rescue work was not hampered.
The cause of the building's collapse was not yet known
In a statement here, Tamil Nadu Chief Minister J. Jayalalithaa said a team from the National Disaster Management Authority has been sent to the accident spot to speed up the rescue efforts.
The reason for the collapse of the building was not clear. While some said the structure was not strong enough, others said the collapse was caused by a lightning strike as it was raining heavily then.
The residential project Trust Heights is being constructed by Prime Sristi Housing Pvt Ltd. The project consisted of two 11-storey blocks - The Belief and The Faith.
Tamil Nadu Governor K Rosaiah expressed his shock over the tragedy.