Four people were confirmed dead and another four, including two children, were hospitalised, officials said, citing the latest information.
National television broadcast footage of rescue workers combing through mangled heaps of concrete and metal in temperatures of minus 18 Celsius.
RIA said it ripped through the building at around 6 a.m. when many residents were asleep.
Emergencies Minister Yevgeny Zinichev said at a meeting with Putin there were "presumably between 36 and 40 people under the rubble" as of Monday evening, agencies reported.
Temperatures in Magnitogorsk were expected to plunge to minus 23 Celsius on New Year's night, the biggest holiday of the year in Russian Federation.
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At least four people were killed and dozens remain unaccounted for after a suspected gas explosion triggered the collapse of a section of a high-rise apartment building in the Russian city of Magnitogorsk early on New Year's Eve. Cabinet officials arrived in Magnitogorsk to oversee the rescue efforts.
Local authorities initially said 68 people registered as residents in the collapsed section of the building were missing, but later tracked down some of them. Residents were evacuated to a nearby school.
Dubrovsky said later that some apartments were leased and in general, 120 people, were housed in the collapsed part of the building.
The cause of the gas explosion is not immediately known.
According to the regional government, Russian Health Minister Veronika Skvortsova and Emergency Situations Minister Yevgeny Zinichev have arrived in Magnitogorsk and are now at the site.
Gas blasts have become a relatively common occurrence in Russian Federation over recent years as the country struggles to cope with aging Soviet-era infrastructure and loose safety requirements, according to Reuters.