The incident occurred at around 1.40 PM when the bus driver on route No 17M from Parrys to Vadapalani tried to take a left turn on the diversion on Anna flyover towards Nungambakkam, police said.
However, they refused to confirm eyewitness accounts that the driver was talking on his mobile phone at the time.
An MTC official said there were 39 persons on the bus, including the driver and conductor.
The incident caused a massive traffic jam, forcing police to divert traffic.
The injured have been admitted to Royapettah Government Hospital and a nearby private hospital, police said.
Health Minister V S Vijay, Social Welfare Minister B Valarmathy, Chennai Mayor Saidai S Duraisamy, City Police Commissioner J K Tripathy, along with top police officers and senior MTC officials visited the spot.
The bus was removed from the spot around three hours later with the help of a crane.
