They include China, France, Ireland and Venezuela. Russia and Norway confirmed they had received applications.
But Russia said Edward Snowden later withdrew the application as the Kremlin had set conditions.
Edward Snowden accuses President Barack Obama of putting pressure on the countries to which he has applied.
The 30-year-old former intelligence systems analyst, who is holed up at Moscow airport, is wanted by the US on charges of leaking secrets.
US Secretary of State John Kerry was expected to discuss Edward Snowden’s case in talks with his Russian counterpart Sergei Lavrov on the sidelines of the ASEAN summit in Brunei.
The WikiLeaks press release said that most of the asylum requests – including to Russia itself – were handed to the Russian consulate at Sheremetyevo airport late on Sunday for delivery to the relevant embassies in Moscow.
The requests were submitted by Sarah Harrison, a British member of the WikiLeaks legal team acting as Edward Snowden’s representative, the statement added.
But Kremlin spokesman Dmitry Peskov later said Edward Snowden withdrew the application to Russia because Moscow had said he should give up “anti-American activity”.
“After learning of Russian’s position yesterday, voiced by President Putin … he abandoned his intention [of staying] and his request to be able to stay in Russia,” he said.
Russian President Vladimir Putin had said that while Moscow “never hands over anybody anywhere”, Edward Snowden could only stay on condition that he stopped damaging Russia’s “American partners” with his leaks.
Edward Snowden has reportedly been in the transit area of Sheremetyevo since arriving there from Hong Kong on June 23.
Dmitry Peskov confirmed he was still there and had not crossed into Russian territory, adding that the former analyst had never been a Russian agent and had never worked with its intelligence services.
Polish Foreign Minister Radoslav Sikorski said on Twitter his country had received an application but that he would not be recommending granting the request.
Norway’s foreign ministry later said its embassy in Moscow had received an application by fax which was “probably from him”.
But Norway, Poland, Germany, Austria, Finland and Switzerland said asylum requests could only be made on their soil.
Venezuelan President Nicolas Maduro, who is on a visit to Moscow, said Caracas had not yet received an asylum application from Edward Snowden but that he had “done something very important for humanity” and “deserved the world’s protection”.
“The world’s conscience should react, the world youth should react, the decent people who want a peaceful world should react, everyone should react and find solidarity with this young man who has denounced and altered the world that they [the US] pretend to control.”
Edward Snowden had previously submitted an application to Ecuador, whose embassy in London is sheltering WikiLeaks founder Julian Assange, and to Iceland.
Ecuador’s President Rafael Correa told the Agence France-Presse news agency on Monday that his country would process Edward Snowden’s asylum request if he managed to enter an Ecuadorean embassy.
However, if he can complete his asylum request on Russian territory, then “the situation can be processed and resolved there,” President Correa added.
Details have also emerged of a letter from Edward Snowden to President Rafael Correa, thanking Ecuador for guaranteeing “my rights would be protected upon departing Hong Kong – I could never have risked travel without that”.
Speaking in Tanzania on Monday, President Barack Obama said Moscow and Washington had held “high level discussions” about Edward Snowden, who he said had travelled to Moscow without valid documents.
Edward Snowden describes himself as “a stateless person”, accusing the US government of stopping him from exercising the “basic right…to seek asylum”.
“The president ordered his vice president to pressure the leaders of nations from which I have requested protection to deny my asylum petitions,” he is quoted by WikiLeaks as saying.
“This kind of deception from a world leader is not justice, and neither is the extralegal penalty of exile. These are the old, bad tools of political aggression. Their purpose is to frighten, not me, but those who would come after me.”
The leaking of thousands of classified intelligence documents has led to revelations that the US is systematically seizing vast amounts of phone and web data.
Asylum requests to 21 countries:
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