Cyclone Michaung will hit the Andhra Pradesh state's coast on Tuesday morning as a "severe cyclonic storm", packing winds of 90-100 kilometres (55-62 miles) per hour winds with gusts up to 110 km/h, the Indian Meteorological Department (IMD) said.
In Tamil Nadu, the airport in state capital Chennai was expected to be closed until late Monday, local media reported, as rains pounded the city.
The IMD issued warnings to those fishing in the area to suspend operations, warning of expected "major damage" to thatch huts along the coast.
Trees were uprooted and vehicles swept away due to the heavy rains, according to images posted on social media.
The police posted images of rescuers evacuating people from waterlogged areas.
Tamil Nadu government official Supriya Sahu warned people not to approach a crocodile, after one was spotted on the streets after being been washed out by the rains.
"No need to panic," Sahu posted on social media.
"There is no possibility of harm to humans IF these animals are left alone and UNPROVOKED."
The cyclone is expected to hit India's southeast coast at the town of Bapatla, on the 300-kilometre (185-mile) long stretch between Nellore and Machilipatnam.
Scientists have warned that storms are becoming more powerful as the world gets warmer with climate change.
Cyclones -- the equivalent of hurricanes in the North Atlantic or typhoons in the Northwest Pacific -- are a regular and deadly menace on coasts in the northern Indian Ocean, where tens of millions of people live.
