The storm "would increase in intensity with gale wind-speed reaching 205-215 kilometres (127-134 miles) per hour at the time of landfall", the Indian Meteorological Department said.
It warned of extensive damage to agricultural crops, homes and power and communication systems.
The army was on standby in both states for emergency and relief operations, the NDTV news network reported.
Choppers and food packages were ready to be dispatched in areas expected to be worst hit by the storm, which was one notch below the most powerful category.
State officials feared the wind speed could touch levels higher than that of a deadly super-cyclone that hit Orissa in 1999, killing nearly 15,000 people.
"We are fighting against nature. We are better prepared this time, we learnt a lot from 1999," the state's Disaster Management Minister Surya Narayan Patra told the NDTV news channel.
Fishermen out at sea along north Andhra Pradesh, Orissa and West Bengal coast were advised to return to coast.
The impact of the weather system was already being felt with heavy rains and gale-force winds on Friday in Orissa and Andhra Pradesh.India and Bangladesh are hit regularly by cyclones that develop in the Bay of Bengal between April and November, causing deaths and widespread damage to property.
Last January, Cyclone Thane struck the southern Indian state of Tamil Nadu, killing 42 people.