Most deaths have taken place in the southern states of Telangana and Andhra Pradesh, where at least 1,118 people have died since last week.
At least 24 people have died from the heat in West Bengal and Orissa.
Temperatures are likely to drop in some parts over the coming days.
Hospitals are on alert to treat heatstroke patients and authorities have advised people to stay indoors.
Heat wave 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, where temperatures climbed to 47C on May 25, 852 people have died.
“The state government has taken up education programs 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 organizations to open up drinking water camps so that water will be readily available for all the people in the towns,” P Tulsi Rani added.
In neighboring Telangana state, 266 people have died in the last week as temperatures hit 118F (48C) over the weekend.
Temperatures fell slightly in Telangana on May 26, and are expected to start dropping in Andhra Pradesh by the end of the week.
The weather is likely to cool further when the summer monsoon begins at the end of the month.
The Indian capital, Delhi, is enduring a week of sweltering heat as the maximum temperature in the city hit a two-year high of 114F (45.5C) on May 25.
The meteorological department has issued a warning for Orissa, Jharkhand and Andhra Pradesh states saying that maximum temperatures there would remain above 113F (45C).
Meteorological officials said the heat wave 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 India.
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