Speaking on a visit to the West African nation of Senegal, Barack Obama said the case would be handled through routine legal channels.
“I am not going to be scrambling jets to get a 29-year-old hacker,” the president added.
Edward Snowden, who faces espionage charges, flew to Moscow last weekend and requested asylum in Ecuador.
Barack Obama said on Thursday that he had not called China and Russia’s presidents about the case, adding: “I shouldn’t have to.”
He told a news conference in the Senegalese capital Dakar: “I’m not going to have one case of a suspect who we’re trying to extradite suddenly being elevated to the point where I’ve got to start doing wheeling and dealing and trading on a whole host of other issues.”
The president added: “My continued expectation is that Russia or other countries that have talked about potentially providing Mr. Snowden asylum recognize that they are a part of an international community and they should be abiding by international law.”
The US has accused Russia and China of helping Edward Snowden, which both deny.
Edward Obama said the leak highlighted significant vulnerabilities at the National Security Agency (NSA), the US electronic spying organization where Edward Snowden worked as a contractor until last month.
“In terms of US interests, the damage was done with respect to the initial leaks,” he said.
Ecuador said on Thursday it had not processed Edward Snowden’s asylum request because he had not reached any of its diplomatic premises.
The country also renounced its multi-million dollar trade relationship with the US, saying its forthcoming renewal would not influence any decision on Edward Snowden’s case.
“Ecuador will not accept pressures or threats from anyone, and it does not traffic in its values or allow them to be subjugated to mercantile interests,” said government spokesman Fernando Alvarado.
He also made an apparently tongue-in-cheek offer of economic aid to the US for human rights training.
The remarks come a day after the chairman of the US Senate foreign relations committee, Robert Menendez, suggested punishing Ecuador economically if it offered asylum to Edward Snowden.
The American is wanted for leaking to media that the US is systematically seizing vast amounts of phone and web data under a surveillance programme known as PRISM.
On Thursday, Beijing accused the US of “double standards” on cybersecurity.
China’s defense ministry said the Prism programme “has revealed the concerned country’s true face and hypocritical behavior”.
Edward Snowden, now 30, fled to Hong Kong on May 20 before flying to Moscow’s Sheremetyevo Airport on Sunday, where Russian authorities say he remains in transit.
Although Russia has no extradition treaty with the US, Washington says it wants Moscow to extradite him without delay.
Russia denies reports its secret police have questioned Edward Snowden, whose US passport has been revoked.
Hong Kong officials said he had been allowed out of the territory because of a mistake in the middle name given on US arrest documents.
The US justice department dismissed that as a “pretext for not acting”.
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