Snowden, wanted by the United States for revealing sensational details of its vast spying operations, flew into Russia from Hong Kong on June 23 and has since languished in the transit zone of the capital's Sheremetyevo airport.
Breaking cover for the first time since he arrived, Snowden, who has no valid travel documents, told a group of human rights activists at the airport on Friday that he was applying for asylum in Russia until he could legally travel on to Latin America.
Meanwhile, the Guardian newspaper journalist who first published Snowden's revelations about the sweeping US surveillance programmes said Snowden possesses data that could prove far more "damaging" to the US government.
Glenn Greenwald told Argentina's La Nacion paper that the 30-year-old had chosen not to release this information.
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"Snowden has enough information to cause more damage to the US government in a minute alone than anyone else has ever had in the history of the United States," Greenwald told the paper. "But that's not his goal."
The condition initially prompted the fugitive to withdraw his asylum application, before Snowden on Friday indicated he did still want refuge in the country.
Russia said it was still waiting Sunday for the expected asylum request, which Snowden had said was to be filed on Friday. It was not clear whether the hold-up was due to the weekend.
The head of Russia's Federal Migration Service (FMS) Konstantin Romodanovsky said Saturday that "there is for the moment no application from E Snowden". If one was made, it would be examined "according to normal legal procedures", he added.