Most deaths have taken place in the southern states of Telangana and Andhra Pradesh, where more than 750 people have died since last week.
Reports say 24 people have died from the heat in West Bengal and Orissa.
Hospitals are on alert to treat heatstroke patients and authorities have advised people to stay indoors.
Heatwave conditions have been prevailing in the two worst-affected southern Indian states since mid-April, but most of the deaths have happened in the past week.
In the worst-hit state of Andhra Pradesh, 551 people have died as temperatures climbed to 47C on Monday.
"The state government has taken up education programmes through television and other media to tell people not to venture into the outside without a cap, to drink water and other measures," news agency AFP quoted P Tulsi Rani, special commissioner for disaster management in the state, as saying.
"We have also requested NGOs and government organisations to open up drinking water camps so that water will be readily available for all the people in the towns," he added.
In neighbouring Telangana state, 215 people have died in the last week as temperatures hit 48C (118F) over the weekend.
The Indian capital, Delhi, is enduring a week of sweltering heat as the maximum temperature in the city hit a two-year high of 45.5C (113.9F) on Monday.
The Hindustan Times newspaper carried a front-page photo of a zebra pedestrian crossing in the city melting in the heat.
"It's baking hot out here - our outing has turned into a nightmare," said Meena Sheshadri, a tourist from the western city of Pune who was visiting a Delhi monument with her children.
"My throat is parched, even though I've been constantly sipping water."
The meteorological department has issued a warning for Orissa, Jharkhand and Andhra Pradesh states saying that maximum temperatures there would remain above 45C (113F).
Meteorological officials said the heatwave was due to a lack of rain.
There are fears that some of the worst-affected states could be hit by drought before the monsoon rains arrive.
The monsoon is expected to hit the southern state of Kerala towards the end of this month before sweeping across the country.