Russian news media had reported that Snowden remained in a transit zone of Moscow's Sheremetyevo Airport, which is separate from the regular departure zones. He has not been seen by any of the journalists that have been roaming the airport in search of him.
Legally, an arriving air passenger only crosses the border after clearing immigration checks.
Russia's Foreign Minister Sergey Lavrov rejected the U.S. push for Snowden's extradition earlier Tuesday, but wouldn't specify his whereabouts, saying only that he hasn't crossed the Russian border.
"We consider the attempts to accuse Russia of violating U.S. laws and even some sort of conspiracy, which on top of all that are accompanied by threats, as absolutely ungrounded and unacceptable," Lavrov said. "There are no legal grounds for such conduct by U.S. officials."
U.S. and Ecuadorean officials had said they believed Snowden was still in Russia.
Kerry called for "calm and reasonableness."
"We would hope that Russia would not side with someone who is `a fugitive' from justice,'" Kerry said at a news conference in Saudi Arabia.
The U.S. has revoked Snowden's passport.
A representative of WikiLeaks has been traveling with Snowden, and the organization is believed to be assisting him in arranging asylum. Assange, the group's founder, said Monday that Snowden was only passing through Russia and had applied for asylum in Ecuador, Iceland and possibly other countries.
A high-ranking Ecuadorean official told The Associated Press that Russia and Ecuador were discussing where Snowden could go, saying the process could take days. He also said Ecuador's ambassador to Moscow had not seen or spoken to Snowden. The official spoke on condition of anonymity because he was not authorized to discuss the case publicly.
Ecuador's foreign minister, Ricardo Patino, hailed Snowden on Monday as "a man attempting to bring light and transparency to facts that affect everyone's fundamental liberties."
"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.
The Kremlin has previously said Russia would be ready to consider Snowden's request for asylum.
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.
Some observers said in addition to the sensitive data, Snowden's revelations have provided the Kremlin with propaganda arguments to counter the U.S. criticism of Russia's crackdown on opposition and civil activists under Putin.
"They would use Snowden to demonstrate that the U.S. government doesn't sympathize with the ideals of freedom of information, conceals key information from the public and stands ready to open criminal proceedings against those who oppose it," Konstantin Remchukov, the editor of independent daily Nezavisimaya Gazeta, said on Ekho Moskvy radio.
Putin has accused the U.S. State Department of instigating protests in Moscow against his re-election for a third term in March and has taken an anti-American posture that plays well with his core support base of industrial workers and state employees.