The storm caused at least five deaths in and around Chennai from trees falling, drowning and electrocution, the Times of Indiareported.
A spokesman for the chief minister's office declined to comment on the reported deaths.
Local administration workers have been working to remove fallen trees and power lines, city corporation officials said on Twitter.
People in Velachery, a low-lying suburb of Chennai, said the storm's impact was mitigated by steps taken by the government ahead of the storm, compared to the hundreds of deaths during floods in 2015.
"The situation was terrible here during the floods in 2015. This year, because of the precautions taken, the situation has not been that bad," said S. Sakthivel, a shopkeeper in Velachery.
More than a hundred cars were parked on the edge of a bridge in Velachery to keep them above the floodwaters, according to a Reuters witness.
While rains are expected to subside in Tamil Nadu, heavy precipitation is expected in the Rayalaseema region of the neighbouring state of Andhra Pradesh on Thursday.
The streets around the Lord Venkateshwara temple, one of the world's richest, in Tirumala, Andhra Pradesh, were flooded and officials were working to drain the water, the New Indian Express newspaper reported.
Tens of thousands of people were evacuated from low-lying areas of Tamil Nadu ahead of the storm's landfall, a state minister said on Wednesday.
The Meteorological Department said Nivar's intensity had dropped to 85 to 95 kph (53 to 59 mph) and is expected to weaken further.