NEW DELHI (Reuters) - India officially declared on Monday that nearly 6,000 people were missing a month after flash floods ravaged large parts of its northern state of Uttarakhand, but stopped short of saying they were presumed dead.

The figure of 5,748, based on tallies of missing persons from around the country, was the first official estimate following weeks in which the numbers of dead and missing fluctuated wildly from a few hundred to several thousand.

Their families will now be eligible for financial relief, Uttarakhand Chief Minister Vijay Bahuguna told a news conference, adding that his government would pay 150,000 rupees ($2,500) to families in the state, besides compensation from the federal government.

"We are not getting into the controversy whether the missing persons are dead or not," said Bahuguna. "We are abiding by what the families of the victims say, and if they think that they haven't come back and have no hope as well, (then) we are providing them monetary relief."

The official death toll still stands at 580, an official of the National Disaster Management Authority told Reuters. More than 4,600 of the missing in Uttarakhand had come from elsewhere in India, said the official, who declined to be identified as he was not authorized to speak to the media.

Record rains in June caused devastating landslides and flooded rivers in Uttarakhand, trapping tens of thousands of Hindu devotees, who flock there each year on a pilgrimage to the temple towns of Kedarnath, Gangotri, Badrinath and Yamunotri.

The rains buried villages in silt and washed away roads, while raging rivers like the Ganges swept away homes on their banks.

The disaster, dubbed a "Himalayan tsunami" by officials and media, prompted one of the largest airlifts in the history of the Indian air force, as helicopters flew hundreds of sorties to rescue residents and pilgrims and drop thousands of kilograms of relief material.

More than 100,000 people were rescued by the air force and security force personnel on the ground, officials said.

($1=59.9250 Indian rupees)

(Refile to fix typo in name of pilgrim town Badrinath in paragraph 6)

(Additional reporting by Nita Bhalla,; Writing by Sruthi Gottipati, Editing by Ross Colvin and Clarence Fernandez)