Snowden has been in the transit zone of Moscow's Sheremetyevo airport for about four weeks now. He applied for temporary asylum in Russia last week.
There were reports that Snowden was given a document allowing him to formally enter Russia while his application was considered. Some Russian news agencies cited unidentified sources saying his lawyer, Anatoly Kucherena, would deliver the documents to Snowden, but Kucherena later said there was no such paperwork.
Venezuela, Bolivia and Nicaragua have offered him asylum, but for now, Russia is Snowden's final destination, Kucherena said. Snowden plans to start studying the Russian language and culture.
The U.S. wants Snowden sent home to face prosecution for espionage after he revealed details of the NSA's spy activities.
President Vladimir Putin has said that Snowden can be granted asylum in Russia if he stops leaking NSA secrets, which has added new tensions to U.S.-Russia relations. It remains unclear how Snowden's status would impact President Barack Obama's plans to travel to Russia in September.
According to the latest ABC News/Washington Post poll, public attitudes have shifted against Snowden, with more than half of Americans supporting criminal charges.
The Associated Press contributed to this report.