WikiLeaks also posted a statement attributed to Snowden on its website late Monday, in which he slams President Barack Obama for "using citizenship as a weapon."
"Although I am convicted of nothing, (the United States) has unilaterally revoked my passport, leaving me a stateless person," Snowden says in the statement. "Without any judicial order, the administration now seeks to stop me exercising a basic right. A right that belongs to everybody. The right to seek asylum.
"Their purpose is to frighten, not me, but those who would come after me."
The Russian government says that Snowden, who has been on the run since releasing the sensitive NSA documents, has remained in the transit zone of Moscow's Sheremetyevo Airport since his arrival from Hong Kong on June 23.
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 in 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.