"We agree with (the Russian) President (Vladimir) Putin that we don't want this issue to negatively impact our relations, and we're simply asking Russia to build on our cooperative history of working together on law enforcement matters and to go ahead and expel Mr Snowden," State Department spokesperson Patrick Ventrell said.
Snowden, who leaked the secretive phone and Internet surveillance programme, is currently stranded in the transit area of the Moscow airport.
"We've had our differences with Ecuador, but we've also found ways to cooperate. So we're making the same case that this is an individual who has been charged with three serious felonies, and he should be returned to the United States and stand trial. So that's the point we're making there," he said.
Granting asylum to Snowden would not be a good thing, he said.
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"What would not be a good thing is them granting Mr Snowden asylum. That would have grave difficulties for our bilateral relationship. That would cause there to be great difficulties in our bilateral relationship. So taking the lens back a little bit, if they take that step, that would have very negative repercussions," Ventrell said.
"I mean, there is not an extradition treaty. We're making clear through bilateral channels our strong desire to have him returned to the United States, and the Russians are very clear about how we feel about Mr Snowden," he said.