"We're following all the appropriate legal channels and working with various other countries to make sure that the rule of law is observed," President Barack Obama told reporters.
Some experts said it was likely that Russian spy agencies were questioning Snowden on what he knows about U.S. electronic espionage against Moscow.
"If Russian special services hadn't shown interest in Snowden, they would have been utterly unprofessional," Igor Korotchenko, a former colonel in Russia's top military command turned security analyst, said on state Rossiya 24 television.
The Kremlin has previously said Russia would be ready to consider Snowden's request for asylum.
The state ITAR-Tass news agency cited unidentified sources as saying that Snowden hasn't applied for a Russian entry visa and can't cross the border without it. It said that he has remained in the transit zone of the Sheremetyevo airport.
Legally, an arriving air passenger "crosses the border" after clearing immigration checks.
The Interfax news agency, which has close contacts with Russian security agencies, quoted an unidentified "well-informed source" in Moscow as saying Tuesday that Snowden could be detained for a check of his papers if he crosses the Russian border. The report could reflect that authorities are searching for a pretext to keep Snowden in Russia.
Snowden is a former CIA employee who later was hired as a contractor for the NSA. In that job, he gained access to documents that he gave to newspapers the Guardian and The Washington Post to expose what he contends are privacy violations by an authoritarian government.
Snowden also told the South China Morning Post newspaper in Hong Kong that "the NSA does all kinds of things like hack Chinese cellphone companies to steal all of your SMS data." He is believed to have more than 200 additional sensitive documents in laptops he is carrying.