Snowden is wanted by the US authorities for leaking sensational details of vast US surveillance programmes.
The Kremlin says he has been in the transit zone of Moscow's Sheremetyevo airport since he arrived on a flight from Hong Kong on Sunday.
But in a mystery that has captivated the world, there has not been a single sighting of him at the airport and his travel plans remain a mystery after he failed to board a flight to Havana on Monday.
Ecuador, seen as his most likely eventual destination, on Thursday denied it had given Snowden a "safe passage" document.
Ecuador's President Rafael Correa told reporters in Quito on Thursday that the government had yet to consider his case.
"We would probably examine it, but for now he is in Russia," he added.
Ecuador's ambassador to Russia had met Snowden just once on Monday in the transit area of Moscow's Sheremetyevo airport, he said.
But the affair has already triggered a war of words between the US and Ecuador.
It also risks further aggravating tensions, already strained over the Syria conflict, between Washington and Moscow, as well as Beijing.
Obama nevertheless made it clear there were limits to what the United States would do to catch Snowden.
"I am not going to be scrambling jets to get a 29-year-old hacker," he said on Thursday during a visit to Senegal, giving the wrong age for the 30-year-old former National Security Agency (NSA) technician.
Obama indicated he did not want to ruin ties with Moscow and Beijing for the sake of Snowden, saying their relationships were broad and ranged over many issues.
He also rejected any "wheeling and dealing" over Snowden.
He had not called President Xi Jinping of China or Russian President Vladimir Putin to discuss the issue, he said.
"The reason is, number one, I shouldn't have to. This is something that routinely is dealt between law enforcement officials in various countries," said Obama.
Putin has indicated that Moscow is keen to see the back of its unexpected visitor, while also strongly rejecting US pressure to hand over Snowden. The two countries have no extradition treaty.
"The sooner this (he flies onwards from Moscow) happens, the better," said Foreign Minister Sergei Lavrov.
On Thursday Ecuador dropped out of a trade pact with the United States, claiming it had become an instrument of "blackmail."
The US responded by telling Quito it wanted to maintain good economic relations but warned that granting asylum to Snowden would have "grave difficulties" for bilateral ties.
Countering claims by WikiLeaks, Ecuador's Political Issues Minister Betty Tola denied that the government had authorised the delivery of any "safe passage or refugee document" to allow Snowden to travel to the country.
Ecuador's London embassy is giving refuge to WikiLeaks founder Julian Assange as he faces extradition to Sweden on allegations of sexual assault.
'Almost certainly grant asylum'
Venezuelan President Nicolas Maduro meanwhile said Caracas would "almost certainly" grant political asylum to Snowden if he applied.
Maduro, who like Ecuador President Rafael Correa is a leftist anti-American populist, is expected in Moscow next week for an energy summit.
Snowden had been expected to leave Moscow on an Aeroflot flight on Monday to Havana, where he could have caught a connection to Quito. But he did not take that flight, nor any others this week.
The next Moscow-Havana flight is on Saturday but there have been no reports he has a booking.
A source quoted by Russia's RIA Novosti state news agency said Snowden cannot currently leave the transit zone.
"Snowden does not have valid documents. He is not flying to Cuba or anywhere else for that matter," the source said.
Another source quoted by Interfax said the situation was at a "dead end" and there would only be a breakthrough when a third country formally offered Snowden asylum.
The confusion has raised the prospect that Snowden could be in limbo for weeks or even months.
Putin has denied suggestions that Russia could be holding up Snowden deliberately to allow an extensive debriefing at the hands of Russian special services.
Snowden abandoned his high-paying intelligence contractor job in Hawaii -- which he himself described as "living in paradise, making a ton of money" -- and went to Hong Kong on May 20.
He then began issuing a series of leaks on the NSA gathering of phone call logs and Internet data, triggering concern from governments around the world.
He arrived in Moscow on Sunday on an Aeroflot flight from Hong Kong, a special administrative region under Chinese rule, prompting anger from Washington that the local authorities there had allowed Snowden to travel.