Snowden, who was hiding out in Hong Kong until Sunday when he left for Moscow, was supposed to have been aboard a flight to Havana Monday, The Guardian of Britain reported. However, a number of journalists who were also on that flight reported Snowden checked in but was not on board for takeoff.
"Mr. Snowden has so far not been seen, but something out of the ordinary is definitely happening, judging by the security and the amount of media," said Russia Today's Egor Piskunov, reporting from the plane.
It had been speculated Snowden was due to fly to Havana and then to Ecuador, where his request for political asylum was being processed. Snowden is trying to evade a U.S. extradition order on theft and spying-related charges.
A member of the airport's security team told ITAR-Tass Snowden hadn't left the airport's transit zone, where passengers await connecting flights to other countries.
"Snowden did not leave on the Aeroflot flight to Havana and is still in the transit zone," the team member said.
Caitlin Hayden, the chief spokeswoman for the U.S. National Security Council, said in a statement Monday Russia allowing Snowden to leave would set back the country's relationship with the United States.
"We now understand Mr. Snowden is on Russian soil," Hayden said. "Given our intensified co-operation after the Boston marathon bombings and our history of working with Russia on law enforcement matters -- including returning numerous high level criminals back to Russia at the request of the Russian government -- we expect the Russian government to look at all options available to expel Mr. Snowden back to the U.S. to face justice for the crimes with which he is charged."
Russian officials said they have no authority to detain Snowden.
"The Americans can't demand anything," human-rights ombudsman Vladimir Lukin told Interfax News Agency.
In Hong Kong, an official with knowledge of government deliberations that took place on whether to allow Snowden to leave Hong Kong said the government of Hong Kong deliberately decided not to stop Snowden from departing, The New York Times reported.
The official, who wished to remain anonymous, said Hong Kong stalled for time by telling the United States its request for Snowden's extradition was incomplete.
U.S. officials maintained the request was complete and included all information needed to arrest and extradite Snowden.
Meanwhile, the U.S. government revoked Snowden's passport, The Washington Post reported. U.S. diplomats and law enforcement officials also have warned countries in Latin America not to harbor Snowden or allow him to pass through to other destinations.
The State Department, which issues and oversees U.S. passports, had no direct comment on the Post report.
But spokeswoman Jennifer Psaki said in a statement, "Persons wanted on felony charges, such as Mr. Snowden, should not be allowed to proceed in any further international travel other than is necessary to return him to the United States."
The State Department may deny or revoke U.S. passports for foreign policy or national security reasons.
The Obama administration charged Snowden with theft of U.S. government property, unauthorized communication of national defense information and willful communication of classified communications intelligence information. The last two charges are espionage-related.
It was not immediately clear if his lack of a valid passport would prevent his reported travel plans. The Post said Snowden's passport was invalidated Saturday, but he still flew on a commercial jet to Moscow from Hong Kong Sunday, even after Washington asked Hong Kong to arrest Snowden in anticipation of extradition.
Whistle-blowing website WikiLeaks said it helped Snowden flee to Moscow and arrange special refugee travel documents from Ecuador.
Ecuador has been helping WikiLeaks founder Julian Assange avoid extradition to Sweden by letting him stay at its embassy in London since June 19, 2012.
Snowden is "bound for the Republic of Ecuador via a safe route for the purposes of asylum," WikiLeaks said in a statement.
Russia's non-governmental Interfax News Agency said Snowden could journey from Moscow without a U.S. passport if the country where he was seeking asylum provided him with travel documents, such as affirmation of refugee status, or even a passport from the destination country.
U.S. Sen. Chuck Schumer, D-N.Y., criticized Russian President Vladimir Putin for "aiding and abetting" Snowden's travel to Moscow.
"That's not how allies should treat one another, and I think it will have serious consequences for the United States-Russia relationship," Schumer told CNN's "State of the Union" Sunday.
Schumer said he also suspected China had a hand in letting Snowden leave Hong Kong -- a suspicion The New York Times Sunday confirmed was true.
Sen. Lindsey Graham, R-S.C., said Snowden's choice of countries for support undercut his claims to be defending privacy and freedom.
"The freedom trail is not exactly China, Russia, Cuba, Venezuela," he told "Fox News Sunday."
"So I hope we'll chase him to the ends of the Earth, bring him to justice and let the Russians know there will be consequences if they harbor this guy," he said.
The Kremlin said it had "nothing to do with" Snowden's activities in Moscow.
"I am not in charge of tickets. I don't approve or disapprove plane tickets. We're not the proper people to address this question to," Putin spokesman Dmitri Peskov told The New York Times.
Ecuadorean Foreign Minister Ricardo Patino said in a Twitter message his government "received an asylum request from Edward Snowden."