Russian media say the migration service issued the document, but the service has not yet commented.
He has stayed in transit since arriving from Hong Kong on 23 June, and recently requested temporary asylum in Russia.
The US authorities want him extradited for leaking details of government surveillance programmes.
Russian President Vladimir Putin has refused to hand him to the US authorities, but said he could stay in Russia only if he stopped leaking.
Mr Snowden's lawyer Anatoly Kucherena arrived at the airport earlier, telling reporters only that he was going to see his client.
Unnamed airport officials later told local media that the lawyer had handed Mr Snowden the documents he needed to leave the airport.
The Interfax news agency reported that the American would be provided with new clothes before leaving the airport.
There was no official comment on news of the pass, which is believed to be an official paper confirming that his asylum application is being considered and allowing him to enter Russian territory.
A number of Latin American states say they are willing to offer Mr Snowden asylum, but he says he first needs Russian asylum in order to be able to travel because the US has cancelled his passport.
The US has charged Mr Snowden with leaking classified information.
Mr Snowden revealed that the National Security Agency (NSA) was collecting Americans' phone records, which civil liberties campaigners say is an unwarranted invasion of privacy.
The White House is urging Congress to reject an attempt to end the practice.
Congressman Justin Amash has introduced an amendment to a defence spending bill that would block funding for the controversial programme. A vote is due shortly.
Other revelations stemming from the leaked documents include allegations that US intelligence agencies have been spying on Latin American politicians and businesses, diplomatic missions and EU offices.
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