"The document does not contain requests about extradition or deportation," said the statement without providing further details.
The letter arrived on Wednesday, the same day Snowden was widely expected to leave the transit zone of Sheremetyevo airport but failed to emerge at the last moment.
Washington wants to prosecute the former National Security Agency contractor on espionage charges for revealing details of a vast spying programme and has repeatedly pressed Moscow to send him back to the United States.
The Kremlin has defied calls to extradite him, saying there was no such agreement between the two countries, but also indicated it does not want to see ties damaged over Snowden.
The now famous fugitive has been marooned in Sheremetyevo airport since arriving from Hong Kong on June 23. He applied for temporary asylum in Russia last week and had been widely expected to be given a pass to enter into Russia proper on Wednesday.
A pro-Kremlin lawyer assisting Snowden with his asylum request, Anatoly Kucherena, told reporters however that Snowden was so far staying at the airport and the legal process was taking longer than expected.