Anatoly Kucherena told reporters his client would remain in the transit zone at Moscow's Sheremetyevo airport, where he has been for the past month.
Earlier, airport officials said that Mr Kucherena had given Mr Snowden the travel documents.
The US wants him extradited for leaking details of surveillance programmes.
Russian President Vladimir Putin has refused to hand him to the American authorities, but said he could stay in Russia only if he stopped leaking US secrets.
Mr Snowden, whose passport has been cancelled by the US, has stayed in the airport's transit area since arriving from Hong Kong on 23 June.
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He has requested temporary asylum in Russia, and said recently his favoured final destination was Latin America.
Mr Kucherena visited his client at the airport on Wednesday and told a news conference later that the 30-year-old would not yet be leaving the airport.
He said the application for temporary asylum had not been processed, and no travel documents had been received.
"He is not planning to leave for now. He asked for temporary asylum, which in the case of a positive decision is granted for a term of one year," said Mr Kucherena.
The lawyer said Mr Snowden wanted to study Russian culture, adding that he had given the American books including Dostoyevsky's Crime and Punishment.
Mr Kucherena said Mr Snowden was "anxious" about leaving the airport.
"I hope that this situation will be resolved in the nearest future," he said. "This is the first time Russia is facing such a situation, and this issue of course requires time for the immigration workers."
The lawyer's account contradicted that of airport officials, who earlier told local media that the lawyer had handed Mr Snowden the documents he needed to leave the airport.
It was unclear whether there had been last-minute political intervention or some other problem.
The Interfax news agency quoted unnamed officials saying the American would be provided with new clothes before leaving the airport.
After the lawyer's news conference, the White House said it wanted clarity from Russia on his status.
Spokesman Jay Carney said Washington had already made it clear to Moscow that Mr Snowden should be handed over to face espionage charges.
The Snowden affair has caused diplomatic ructions around the world, upsetting America's close allies and traditional enemies.
Leaks by the former CIA worker have led to revelations that the US National Security Agency (NSA) is collecting Americans' phone records.
On Wednesday evening, an attempt to block funding for the programme narrowly failed in a 205-217 vote in the US House of Representatives.
The White House had lobbied Congress to support the surveillance.
Opponents of the US, including Bolivia, Venezuela and Nicaragua, have all offered Mr Snowden asylum.
Fugitive US intelligence leaker Edward Snowden has not been given Russian travel documents, his lawyer has said, contradicting earlier reports.