Reports say the 1985-built Kolskaya may have sank after being battered by a fierce storm in the Sea of Okhotsk off Sakhalin Island at 14:00 local time.
The 67 crew members on board the rig, which was under tow from a contract for Gazflot off West Kamchatka to Sakhalin when it capsized, were awaiting evacuation by helicopter when disaster struck.
Large waves and iced-over portholes are initially being blamed for the incident as water filled the vessel.
Fourteen people have been rescued and a small number of bodies found, but 49 people remain missing, according to Russia’s Emergencies Ministry.
Attempts to locate the missing crew members have been hampered by severe weather conditions.
The rig is owned by state drilling company, ArktikMorNefteGazRazvedka, which is based in Murmansk.
In April, Gazprom offshore affiliate contracted it to drill an appraisal well this summer on the West Kamchatka block.
Gazprom, which received a licence for the West Kamchatka block in 2009, has said earlier that it was planning to drill two appraisal wells on the acreage this year. In total, four wells were to be drilled on four perspective structures under the licence terms.
However, drilling of the appraisal well on the West Kamchatka block had been halted in September after Russian environmentalists discovered it had not been given approval.