Snowden, the target of a US arrest warrant issued Friday after he blew the lid on massive secret surveillance programmes, arrived in Moscow on a direct flight operated by Russian flag carrier Aeroflot.
The Hong Kong government said earlier it had "no legal basis" to prevent Snowden leaving because the US government had failed to provide enough information to justify its provisional arrest warrant for the former National Security Agency (NSA) contractor.
Russian media reports citing sources within Aeroflot said he would fly to Cuba tommorow and then board a flight to the Venezuelan capital Caracas.
"Russian law enforcement agencies have nothing against him and we have no orders to detain him," one law enforcement source told the state news agency Ria Novosti.
Relations between Russia and the United States remain frosty because of discord over a raft of issues including the Syria conflict, reflected in a tense meeting between Presidents Barack Obama and Vladimir Putin at the G8 summit in Northern Ireland last week.