Arrival of flights in Chennai has been suspended till evening in view of showers and heavy winds though departures continued.
India's northeast monsoon usually runs from October to December, bringing heavy rain, particularly to the south.
The Indian Meteorological Department forecast light to moderate rain in most affected areas, with occasional flooding of some roads and low-lying areas.
Many parts of the state capital Chennai, which is India's auto manufacturing centre, were waterlogged and government officials used pumps to drain some communities that had been stranded waist deep.
Many schools and colleges in Tamil Nadu remained closed while some train services were still suspended.
The depression, that moved towards the metropolis on Thursday is all set to cross the coast here soon while the authorities used heavy duty motors, robotic excavators and deployed its full manpower to pump out water, de-clog drains, remove trash and uprooted trees.
Chief Minister M K Stalin directed officials specially deputed to manage rain-related issues in various districts to expedite the relief activities and ensure quality food and medical facilities in relief camps.
Flooding in Chennai killed more than 250 people during record rainfall in 2015.
Scientists say ever-more unpredictable and extreme weather across South Asia has been caused by climate change and exacerbated by deforestation, damming and excessive development.