Hundreds of trees have been uprooted and power lines brought down in Andhra Pradesh and Orissa states.
Three people have so far been reported killed in Andhra Pradesh.
It is feared a storm surge of up to two metres could inundate low-lying areas and hundreds of relief centres have been opened in the two states. Disaster relief teams have also been sent.
The authorities say the next five to six hours will be crucial.
The strength of the cyclone has been revised upwards since Friday, and the Indian Navy is on standby to assist.