AFP reports citing Wikileaks.
In addition to earlier requests to Ecuador and Iceland, Wikilieaks said its legal advisor in the Snowden matter, Sarah Harrison, had personally submitted other requests on behalf of Snowden, the NSA whistleblower.
"The requests were delivered to an official at the Russian consulate at Sheremetyevo Airport in Moscow late in the evening," a WikiLeaks statement posted online said.
"The documents outline the risks of persecution Mr Snowden faces in the United States and have started to be delivered by the Russian consulate to the relevant embassies in Moscow."
The petitions on Snowden's behalf were made to Austria; Bolivia; Brazil, China, Cuba, Finland, France, Germany, India, Italy, Ireland, Netherlands, Nicaragua, Norway, Poland, Russia, Spain, Switzerland and Venezuela, the group added.
Snowden on Monday accused US President Barack Obama of "pressuring the leaders" of countries from which he has sought protection.
In his first public announcement since fleeing Hong Kong eight days ago, Snowden accused Obama of having ordered Vice President Joe Biden to put pressure on leaders of countries where he was seeking asylum.
It also emerged that Snowden had written to Ecuador's President Rafael Correa thanking him for his support in his bid to avoid extradition.
"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," he wrote in the letter, obtained by Britain's Press Association.
Correa has said Biden raised the issue of Snowden in a conversation over the weekend, asking him to reject the fugitive computer analyst's asylum request.
Snowden, in his statement issued from his refuge at Moscow's Sheremetyevo airport, said Obama was guilty of "deception" and imposing "the extralegal penalty of exile.