News agency Interfax cited Kim Shevchenko, the duty officer at the Russian Foreign Ministry's consular office in Moscow's Sheremetyevo airport, as saying that Snowden's representative handed over his request Sunday.
Snowden has been caught in legal limbo in Moscow since he arrived there from Hong Kong on June 23. His passport has been annulled by the U.S., and Ecuador has been coy about offering him shelter.
Russian President Vladimir Putin says Snowden will have to stop leaking U.S. secrets if he wants to get asylum in the country. However Putin added that Snowden has no plan to quit doing so.
The Associated Press contributed to this report.