The biggest hurdle is the power and influence of the United States. Because Snowden's U.S. passport has been revoked, the logistics of him departing are complicated.
Venezuela, Nicaragua and Bolivia have made asylum offers over the past two days, but the countries haven't indicated they would help the fugitive by issuing a travel document, which he would need to leave Russia.
The former NSA systems analyst is charged with violating U.S. espionage laws. He is believed to be stuck in the transit area of Moscow's airport after arriving from Hong Kong June 23.
Russia doesn't appear to be willing to help Snowden leave the airport, with a spokesman saying the issue of his travel documents is "not our business."
President Vladimir Putin said earlier that Snowden would be offered asylum in Russia if he stopped leaking U.S. secrets. Snowden then withdrew his Russian asylum bid, a Russian official said.
President Barack Obama has publicly displayed a relaxed attitude toward Snowden's movements, saying in June that he wouldn't be "scrambling jets" to capture him. However, other senior U.S. officials have used harsh language saying they want him back.
White House spokesman Jay Carney said China had "unquestionably" damaged its relationship with Washington for not returning Snowden, who recently turned 30, from semi-autonomous Hong Kong while he was still there.
The asylum offers from Venezuela, Nicaragua and Bolivia came after leftist South American leaders denounced the rerouting of Bolivian President Evo Morales' plane over Europe amid reports that Snowden was aboard.
Spain says it had been warned along with other European countries that Snowden was aboard the Bolivian presidential plane, an acknowledgement the manhunt for the fugitive leaker had something to do with the plane's unexpected diversion to Austria.
It is unclear whether Washington warned Madrid about the Bolivian president's plane. On Saturday, Morales offered asylum to Snowden, but didn't say if Bolivia had received a request from him.
U.S. officials have declined to comment on the grounding of Morales' plane, except to say that they won't give details about their conversations with European countries and that they have already stated Washington's position that it wants Snowden back.
The Associated Press contributed to this report.