Russian attorney Anatoly Kucherena said he saw Snowden write the request and submit it to a Federal Migration Service representative at Sheremetyevo International Airport, where Snowden has been holed up since arriving in Russia from Hong Kong June 23, RIA Novosti reported.
WikiLeaks, a whistle-blowing group that publishes leaked secret information through its website, posted on its Twitter page Tuesday that Snowden applied for a "temporary protection visa" in Russia, CNN reported.
Snowden needs temporary legal status in Russia to secure eventual passage to any of the Latin American countries that have offered him asylum.
The U.S. State Department revoked Snowden's passport as part of Washington's effort to get Snowden returned to the United States, where he faces several charges, including two espionage-related charges, after he leaked information about National Security Agency surveillance programs.
Documents Snowden leaked revealed U.S. National Security Agency domestic and international spying, including logging Americans' phone calls and intercepting foreigners' Internet communications, with help from U.S. companies, through a program called PRISM.
Russian President Vladimir Putin said last month Moscow would consider granting Snowden asylum provided the former NSA contractor cease his work aimed at "damaging our American partners," an option Snowden at first rejected but later reconsidered.
Snowden left Hawaii for Hong Kong earlier this year and leaked documents that exposed the NSA's massive monitoring programs. After he admitted he was the leaker in June, Snowden fled Hong Kong for Russia.
Putin said Monday Washington trapped Snowden in Russia by frightening leaders who might otherwise take him.
He also said he hoped Snowden would find a place to go and leave.
"He arrived on our territory without an invitation," Putin told student researchers attending an archaeological camp on Russia's Hogland Island in the Gulf of Finland.
"He didn't fly to us -- he flew in transit to other countries. But only when it became known that he was in the air, our American partners, in fact, blocked him from flying further," Putin said.
"They themselves scared all other countries. No one wants to take him, and in this way they themselves in fact blocked him on our territory. Such a present for us for Christmas," Putin told the students.
Putin later told reporters, "As soon as there is an opportunity for him to move elsewhere, I hope he will do that."