Snowden made his first public statement in nearly three weeks last Friday as he met with a group of Russian rights activists and pro-Kremlin figures in the Moscow airport where he had been marooned since June 23 after flying in from Hong Kong.
However he has made no further public statement and has not yet filed the asylum application with the Russian authorities that he told the activists he would be making Friday evening.
The head of Russia's Federal Migration Service (FMS), Konstantin Romodanovsky, told Russian news agencies Monday that no such application had come in yet.
"There have been no applications from Edward Snowden as of today," he told Interfax.
A source in Sheremetyevo airport said that Snowden still has a room reserved in the capsule hotel in the transit zone, but that he also has access to a "special rest room for staff on duty," Interfax reported.
"Snowden's safety is being guarded both in the hotel and in this room," the source said, without elaborating whether such precautions were handled by a security firm or a government agency.
Russian President Vladimir Putin said earlier this month that Snowden could claim asylum in Russia only if he stopped harming US interests.
The condition initially prompted the fugitive to withdraw his asylum application, but Snowden on Friday indicated he did still want refuge in the country due to his inability to travel on.
A journalist who published a series of stories based on Snowden's leaks in the Guardian newspaper Glenn Greenwald said over the weekend that the former NSA contractor has enough information to damage the US government "than anyone else has ever had in the history of the United States" but chose not to publish it.
Washington has reacted sharply to the possibility that Moscow might offer Snowden a safe harbour.
"Providing a propaganda platform for Mr Snowden runs counter to the Russian government's previous declarations of Russia's neutrality," said White House spokesman Jay Carney.
US President Barack Obama spoke to Putin by telephone Friday on issues including the Snowden affair, the Kremlin and White House both said, but no details were released.