Reports indicated that Russian authorities were preparing to let Snowden leave the transit zone of Sheremetyevo airport but then - in mysterious circumstances - he failed to emerge.
Snowden has applied for asylum in Russia and is now awaiting a document that would allow him to formally cross the border and move freely in Russia while his application is being considered - a prospect the United States has said would be "deeply disappointing".
Russian Foreign Minister Sergei Lavrov spoke with US Secretary of State John Kerry about Snowden's case Wednesday but it was not clear if their discussion had an impact on the fugitive's fate.
Washington wants to put the 30-year-old former National Security Agency contractor on trial for revealing details of vast US surveillance programmes, but Moscow has rejected demands to hand him over.
The US ambassador in Russia on Thursday reiterated that Washington wants Moscow to hand over Snowden, despite the absence of an extradition agreement between the two countries.
"The US is not asking for 'extradition', but simply the return of Mr Snowden. We have sent many people back to Russia," Ambassador Michael McFaul wrote on Twitter on Thursday morning.
The Russian foreign ministry made no mention of Kerry and Lavrov discussing Snowden's fate, merely saying on its website that the two diplomats spoke on bilateral relations and the situation in Syria.
But US State Department spokeswoman Jen Psaki told journalists that Kerry called Lavrov after learning that Snowden was to receive a temporary document allowing him to leave the transit zone.
"Any move that would allow him to depart the airport would be deeply disappointing," she said. "Our belief is the only place he should be moving is back to the United States."
Snowden is believed to have stayed in the airport since June 23, after arriving there from Hong Kong.
'He hasn't been given the document'
A flurry of activity at the Moscow airport Wednesday followed reports on Russian news agencies that the US fugitive had received a note from the migration service that his application is being processed, legally allowing him to exit the transit zone.
A strong police presence in the terminal where he is believed to be staying also led people to believe that Snowden was about to formally cross the border after a month in airport limbo.
However, these expectations were suddenly squashed by pro-Kremlin lawyer Anatoly Kucherena, who met with Snowden Wednesday, bringing him some clothing and books, and then facing the media alone.
"As of today, this document (allowing him to leave) has not yet been given (to him)," he told the throng of waiting reporters.
There was no explanation for why the plan appeared to have been thwarted at the last minute, and whether pressure from Washington somehow kept Snowden in the airport.
Full processing of an application for temporary asylum, which Snowden requested on July 16, takes up to three months, authorities have said.
Snowden met with a group of 13 rights activists, lawyers, and lawmakers on July 12. He has shied away from the press and asked not to be photographed due to safety concerns.
The Kremlin has sought to keep Snowden's case at arm's length, but observers said the activists' meeting at the state-controlled airport would not been possible without government involvement.
Kucherena said Snowden wants to stay in Russia and may even apply for citizenship and seek a local job.
With the help of Sarah Harrison, an employee of anti-secrecy group WikiLeaks, the 30-year-old applied for asylum in 27 countries, but most denied him a safe haven under pressure from Washington.
Three countries in Latin America known for their anti-US stance - Bolivia, Nicaragua and Venezuela - have said they would be willing to give him refuge.