The newspaper's source says the sect members managed to put down the fire with difficulty and refused help offered by police.
The officers had long ago installed microphones into the underground bunker in Russia's Penza region, where around 30 people with small children are waiting for the end of the world they say will come in spring.
The police were alerted to the fire when they heard shouts in the bunker. They called emergency services but the sect members did not agree to any medical treatment.
Locked in the bunker since mi-November, they have repeatedly refused to leave or let the children out.
They have vowed to blow themselves up if any attempts to free them are made.