Most of the main streets are waterlogged, bringing the city to a standstill.
Many offices, schools and colleges have been shut due to rains in the past week.
The rains have also affected at least 80,000 people in northern parts of neighbouring Sri Lanka.
A depression formed in the Bay of Bengal has triggered rains in coastal areas of India's Tamil Nadu state and Sri Lanka.
Government officials said around 10,000 people had been evacuated from their homes in Chennai, the Reuters news agency said.
The National Disaster Response Force and the army are also helping in ongoing rescue work.
The Chief Minister of Tamil Nadu, J Jayalalitha, has announced a 5bn-rupee ($75m; £49m) relief fund for flood-affected areas.
"The rain that was meant to be spread out over the monsoon months has poured in just a few days," she said.
"No precautionary measures would have managed to prevent water logging and damages. In areas where flooding and damage have been caused, relief, rescue and repair works are being taken up on a war footing," she added.
Images from Chennai show the difficulties people have been facing to access essential services.