Ecuador, where he had initially hoped to get asylum, has been giving mixed signals about offering him shelter.
Britain's Press Association news agency said it had obtained a letter from Snowden to Ecuadorean President Rafael Correa thanking him for considering his asylum request.
"There are few world leaders who would risk standing for the human rights of an individual against the most powerful government on earth, and the bravery of Ecuador and its people is an example to the world," PA quoted the letter as saying. The agency said it had obtained the Spanish-language letter from sources in Quito, the capital of Ecuador.
Correa, however, appeared cool to Snowden an interview with the Guardian newspaper.
Asked whether he would like to meet Snowden, Correa was quoted as saying: "Not particularly. He's a very complicated person. Strictly speaking, Mr. Snowden spied for some time."
He was quoted as saying that Ecuador would not consider an asylum request until Snowden was on its territory and his government would not help him travel to Ecuador.
The expanded requests for asylum come as the Obama administration contends with European allies angry about the release of documents that alleged U.S. eavesdropping on European Union diplomats.
Obama said Monday that the U.S. would provide allies with information about new reports that the NSA had bugged EU offices in Washington, New York and Brussels. But he also suggested such activity by governments would hardly be unusual.
French President Francois Hollande said Tuesday that France has "not received any particular requests from Mr. Snowden." He also called for a common EU stance on the NSA snooping.
France prides itself on being a haven for political prisoners, and offers asylum to more people annually than any other country but the United States. French politicians on the far right and far left have called for France to take Snowden in — as have members of the Green party, which is a part of Hollande's Socialist-led government. However, it is unlikely that France would take in Snowden. Hollande and leading French officials, despite outraged comments in recent days, sees the US as a key ally.
A foreign ministry spokeswoman in Beijing said it was not aware of Snowden's possible plea to seek asylum in China.
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Associated Press writers Lynn Berry in Moscow, Frank Jordans in Berlin, George Jahn in Vienna, Matti Huutanen in Helsinki, Finland, Monika Scislowska in Warsaw, Poland, Angela Charlton in Paris, Ciaran Giles in Madrid, Spain, and AP researchers Zhao Liang and Yu Bing in Beijing contributed to this report.