The anti-secrecy group which has adopted Snowden and his cause said on its website that WikiLeaks legal adviser Sarah Harrison delivered the requests for asylum to an official at the Russian consulate at the Moscow airport on Sunday.
The WikiLeaks statement says requests were made to China, Cuba, Nicaragua, Venezuela, India, and several European countries.
A Russian government spokesperson told the Reuters news agency that Snowden, who is believed to be holed up in the transit area of Moscow's Sheremetyevo airport, withdrew his request for asylum in Russia after President Vladimir Putin said he can stay if he promises not to harm US interests.
Venezuela said it had not yet received Snowden's request but President Nicolas Maduro said the whistleblower deserved the "world's protection".
Meanwhile, India, Norway, Finland, Austria, and Poland confirmed that they had received Snowden's request on Tuesday.
But Poland's foreign minister said he would not recommend granting a request.
"We received a document that does not meet the requirements for a formal application for asylum," Radoslaw Sikorski wrote on his Twitter account. "Even if it did, I will not give a positive recommendation."
A Finnish foreign ministry spokeswoman said Finland could not accept Snowden's application as Finnish law required him to be in the country.
Austrian Interior Minister Johanna Mikl-Leitner told the Austria Press Agency that Snowden would need to submit such a request directly in Austria but would not be deported if he arrived in Austria because "there is no international arrest warrant".
A spokesperson for India's foreign ministry said the Indian Embassy in Moscow did receive the application but "see no reason to accede to the Snowden request" after careful consideration.