A fire broke out about 1.30am in a bar called Le Constellation in the resort in southwestern Switzerland.
The cause of the blaze, which was initially reported as an explosion, remains unclear but authorities said it appeared to be an accident.
"At the moment we are considering this a fire and we are not considering the possibility of an attack," prosecutor Beatrice Pilloud told a press conference, adding that authorities had opened a full investigation.
Some of the victims are from other countries, said Stephane Ganzer, head of security for the Valais canton.
Two French women who said they were in the bar told France's BFM TV that they saw the fire start in the basement section of the club after a bottle containing "birthday candles" was held up too close to the wooden ceiling.
"The fire spread across the ceiling super quickly," one of the two women, who identified themselves as Emma and Albane, told BFM TV.
The pair said they were able to climb a narrow staircase to the ground floor as panic broke and escape the building.
Minutes later, the fire had reached the ground floor too, they said.
It was not immediately possible to verify their account from other sources.
Video footage, verified by Reuters, showed fire spreading from the building with people outside the club, some running and screaming.
The canton's head of police, Frederic Gisler, said a helpline had been opened for relatives.
"I can't hide from you that we are all shaken by what happened overnight in Crans," Gisler told the press conference.
"Our count is about 100 injured, most seriously, and unfortunately tens of people are presumed dead," he said, adding that patients had been dispatched to hospitals in Sion, Lausanne, Geneva and Zurich.
The Italian foreign ministry said information from Swiss police indicated about 40 deaths but Gisler declined to give a specific figure.
Earlier, police said many people were being treated for burns and that the area had been completely closed off, with a no-fly zone imposed over Crans-Montana.
Authorities said 10 helicopters and 40 ambulances had been deployed.
"What was meant to be a moment of joy turned, on the first day of the year in Crans-Montana, into mourning that touches the entire country and far beyond," Swiss President Guy Parmelin said on X, expressing condolences.
Prosecutor Pilloud said authorities were trying to get the bodies of the victims to their families.
"A lot of resources have been put into forensics to identify the victims. These resources are intended to allow us to get the bodies to the families as soon as possible," she said.
Italian Foreign Minister Antonio Tajani said the fire may have been caused by fireworks.
"It seems to have been an accident caused by a fire, by some explosion, by some firecracker thrown during New Year's celebrations," he told Italy's Sky TG24 tv channel.
