An airport source said Snowden, who is wanted by the United States for revealing details of government intelligence programmes, was handed documents by his lawyer which were expected to include a pass to leave the transit area.
But Snowden did not go through passport control and lawyer Anatoly Kucherena, who is assisting him with his request for temporary asylum in Russia until he can reach a state that will shelter him, said the American did not have the pass he needed.
It was not clear whether there had been a last-minute political intervention or hitch, or the pass had never been in his possession.
But Kucherena said he hoped Snowden's status would be resolved soon.
In Washington, the White House said it was seeking clarification of his status. "I must say he is of course anxious about it and I hope that this situation will be resolved in the nearest future," Kucherena said at Sheremetyevo.
"This is the first time Russia is facing such a situation, and this issue of course requires time for the immigration workers."
Bolivia, Nicaragua and Venezuela have said they could offer sanctuary to Snowden, who arrived on June 23 from Hong Kong, where he had fled to escape capture and trial in the United States on espionage charges. But none of the three Latin American countries can be reached by a direct commercial flight so Snowden has requested temporary asylum in Russia until he believes he can safely reach one of them.
The United States wants him extradited to face prosecution and has revoked his passport. But Russia has refused to send him home and risks damage to relations with the United States if it grants him temporary asylum – a process which could take three months.
Kucherena confirmed Snowden was staying somewhere in the many corridors and rooms of the transit area between the runway and passport control – an area which Russia considers neutral territory – and that he had learned the Russian translation for "Hi", "Bye-bye" and "I'll ring you".
The 30-year-old had received calls from across Russia, with offers to give him money and a place to stay, and even a suggestion by one woman to adopt him.
He said he had enough money to get by for now. Kucherena said he had brought him fresh underwear and shirts and added that he had given him the novel Crime and Punishment by 19th-century writer Fyodor Dostoevsky and short stories by Anton Chekhov.
President Vladimir Putin signalled last week that he did not want the dispute to derail Russia's relations with the United States, and the decision on temporary asylum could be delayed until after US President Barack Obama visits Moscow for a summit in early September.
Allowing him to stay in Russia even temporarily would upset Washington. It will be Putin's first summit with Obama since the former KGB spy started a new term last year, and precedes a subsequent G20 summit in St Petersburg.
But a refusal would open Putin to criticism at home that he gave into Moscow's former cold war enemy. Both countries have signalled they want to improve ties, strained by issues ranging from the Syrian conflict to Putin's treatment of opponents and Western-funded nongovernmental organisations since he started a third term in 2012.
Putin has said Snowden must stop anti-US activities. Snowden has said he does not regard his activities as hostile to the United States but Kucherena said last week that he had agreed to halt such actions.
Snowden, who has been assisted by the WikiLeaks anti-secrecy group, has not been seen in public since June 23 although he had a meeting at the airport with human rights groups on July 13.
He fears the United States will persuade its allies to prevent him using their airspace, or that his plane might be forced down so that he can be taken into custody and extradited.
Kucherena said earlier this week that he did not rule out Snowden seeking Russian citizenship. There has already been diplomatic fallout from Snowden's leaks, which included information that the US National Security Agency (NSA) bugged European Union offices and gained access to EU internal computer networks, although the Union is an ally.
China, Brazil and France have also voiced concern over the spying programme. US relations with Latin American states have been clouded by the refusal of four US allies in Europe to let a plane carrying Bolivia's president home from Moscow use their airspace.
US lawmakers were also clashing over the case as the House of Representatives debated the 2014 defence spending bill.
Michigan Republican Justin Amash has proposed an amendment that would bar the NSA from collecting telephone call records and other data from people in the United States not specifically under investigation.
Obama opposed Amash's amendment, saying it would "hastily dismantle one of our intelligence community's counterterrorism tools". – Reuters