Videos on social media show passengers using emergency exit slides to escape the burning Aeroflot aircraft.
Two children and a flight attendant were among the dead, Russian media reports.
One witness though said it was a "miracle" anyone escaped the jet, which was reportedly carrying 78 passengers.
It remains unclear what caused the large blaze and emergency landing.
The aircraft is a Sukhoi Superjet-100, and was scheduled to depart at 17:50 local time (14:50 GMT) for the city of Murmansk.
The crew issued a distress signal shortly after departure, Interfax reports.
Reports also suggest it did not succeed in its first emergency landing attempt.
Data from tracking website Flightradar24 appears to confirm reports the emergency landing happened about 30 minutes after take-off.
Aeroflot, Russia's national carrier, said the plane was forced to return to the airport "for technical reasons", but did not elaborate.
In a brief statement, the firm said the plane's engines were burning after landing, but did not clarify the exact sequence of events.
Interfax quotes a source as saying the plane's engines caught fire on the runway after a hard landing, not in mid-air as initially reported.