Most of those succumbing to the soaring temperatures have been in southern Andhra Pradesh state, where at least 900 deaths have been reported.
State-run All India Radio said sunstroke and dehydration had claimed another 100 lives in the northern and eastern parts of India, bringing the nationwide toll to about 1,000.
Meteorologists predicted no relief from the heat wave over the next two days, with monsoon rains in India delayed by a week and likely to hit the southern region this weekend.
In the worst-hit Nalgonda district of Andhra Pradesh state, temperatures soared to 120 degrees in some places, said top district administrator Ram Prakash Sisodia. At least 204 deaths were reported in Nalgonda, he said.
"We are carrying out a publicity campaign to educate people about do's and don'ts to protect themselves from the heat wave," Sisodia said. "We have asked them not to come out into the open between 10 a.m. and 5 p.m."
"People have been asked to keep their heads and ears covered with cloth and not to expose themselves to the heat," he said.
The lack of monsoon rains has compounded the crisis.
On Sunday, some 100,000 Muslims offered special prayers for rain in Hyderabad, the capital of Andhra Pradesh state, following a tradition set by the Prophet Mohammed, who is believed to have led similar prayers for rain more than 14 centuries ago.
In another part of the city, more than 5,000 Hindus gathered on the bed of a dried out lake, chanting sacred hymns and prayers from ancient Hindu texts for a reprieve from the blazing heat.
Sunday was the hottest day for five years in Hyderabad, with the temperature rising to 111 F, said C.V. Bhadram, director of the state Meteorological Center.
Two reservoirs which are the main source of Hyderabad's water supply have dried up and water from nearby towns is being supplied to the city's 5 million people on alternate days.
Yet, amid the misery, people are hopeful.
On Sunday, ornithologists sighted a cuckoo, a tiny bird believed to be a harbinger of the monsoons, according to the Bird Watchers' Society of Andhra Pradesh.
The pied-crested and long-tailed cuckoo migrates from East Africa to India every year just before the onset of the monsoons. It was spotted in a park near the Hussain Sagar lake in Hyderabad, said Siraj Ahmed Taher, the society's president.