“On the first of August, he received a three-year residence permit,” attorney Anatoly Kucherena said.
"He will be able to travel freely within the country and go abroad. He'll be able to stay abroad for not longer than three months," Kucherena said.
Snowden, he said, plans to hold a news conference in Russia soon.
White House press secretary Josh Earnest said Thursday that the administration's position on Snowden "hasn't changed" after reports of the Russian offer.
"He is accused of leaking classified information and faces felony charges here in the United States," Earnest said. "That’s why we believe that he should return to the United States as soon as possible, where he’ll be afforded full due process and the protections that are allowed to U.S. citizens under the law."
But the White House spokesman looked to divorce the Snowden situation from U.S. efforts to sanction Russia over its incursion into Ukraine.
"It’s important for people to understand — and I think this has been made crystal clear to President Putin — that the sanctions are related to the destabilizing actions that Russia has taken in Ukraine and along the border with Ukraine," Earnest said.
Just a week ago, it was unclear whether Snowden would be granted an extension to his asylum. Russia’s government granted him one-year asylum in August 2013, which expired on July 31. RT reported in early July that Snowden applied to extend his stay.
Snowden first arrived in Moscow in late June 2013, but became stranded in an airport for more than a month after flying there from Hong Kong.
He is accused of stealing and leaking highly classified U.S. documents from the NSA for which he faces espionage charges.
This story was updated at 1:59 p.m.
Justin Sink contributed.