Many of the victims were burned or suffocated to death in their cells at the jail in Comayagua late on Tuesday night.
Families flocked to the site, desperate for news. Some prisoners escaped the blaze by breaking through the roof to jump from the building, they said.
Officials are investigating whether an electrical fault caused the blaze.
Dozens of prisoners died trapped in their cells and were burned beyond recognition.
Comayagua firefighters’ spokesman Josue Garcia said there were “hellish” scenes at the prison and that desperate inmates had rioted in a bid to escape the flames.
“We couldn’t get them out because we didn’t have the keys and couldn’t find the guards who had them,” Josua Garcia said.
Lucy Marder, who heads the forensic services in Comayagua, said that reports indicated at least 272 people had been killed.
She later said that 356 people on the prison roster were unaccounted for.
“The majority could be dead, though others could have suffered burns, escaped or survived,” Lucy Marder said.
The prison in Comayagua, some 100km (60 miles) north of the capital, Tegucigalpa, was holding more than 800 inmates.
It was feared many of them had fled in the chaos, officials said.
“I’m looking for my brother. We don’t know what’s happened to him and they won’t let us in,” Arlen Gomez told Honduran radio.
Local hospitals are treating dozens of people for burns and other injuries.
Some of the injured have been taken to Tegucigalpa for treatment, among them 30 people with severe burns.
Firefighters said they had struggled to enter the prison because shots had been fired.
Honduran media reported that there had been a riot in the prison before the fire broke out.
Prison service head Daniel Orellana denied this.
“We have two hypotheses. One is that a prisoner set fire to a mattress and the other one is that there was a short-circuit in the electrical system,” he was quoted as saying by Reuters.
Prisons in Honduras, a country which has the world’s highest murder rate, are often seriously overcrowded and hold many gang members.
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