Snowden, who embarrassed US President Barack Obama's administration with his revelations of massive surveillance programmes, failed to appear on a flight he had apparently booked from Moscow to Cuba.
Russia's Interfax news agency, known for its strong security contacts, said he was likely already out of the country.
US Secretary of State John Kerry meanwhile dubbed Snowden a traitor to his country and warned both Russia and China that their relations with the US might be damaged by their refusal to extradite him.
He was said by Russian officials to have spent the night in Moscow's Sheremetyevo airport awaiting his onward connection.
He had been expected to take Aeroflot's 1005 GMT flight today from Moscow to Havana after airline sources confirmed he had checked in.
But the flight left the terminal at Moscow Sheremetyevo airport -- with a pack of hopeful journalists on board -- and then took off with no sign that the former National Security Agency (NSA) contractor was among the passengers.
The Interfax news agency quoted a Russian security source and an Aeroflot source saying that he was not on board the flight to Havana.
It quoted another source familiar with the matter as saying: "Snowden, most likely, has already left the Russian Federation. He could have left by a different plane.
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