Snowden, accused of leaking National Security Agency secrecy and wanted in the United States for espionage, is believed to be staying in a hotel inside the international transit zone in Sheremetyevo Airport in Moscow after leaving Hong Kong in June.
Snowden applied for temporary asylum weeks after arriving at the airport. Snowden is technically not in Russian territory while he stays in the international area.
Anatoly Kucherena, Snowden's Russian attorney, told the news agency Interfax he believes that Moscow could decide within a week whether to grant Snowden asylum.
"The question of giving him temporary asylum won't take more than a week. I think that in the near future he will have the possibility to leave the Sheremetyevo transit zone," Kucherena was quoted as saying by Interfax.
Russian President Vladimir Putin said Wednesday that Snowden has been warned against taking any actions that would damage relations between Moscow and Washington.
On a visit to the Siberian city of Chita, Putin said "we have warned Mr. Snowden that any actions by him connected with harming Russian-American relations are unacceptable," according to Russian news agencies.
In the U.S. Wednesday, NSA Deputy Director John Inglis told congressional representatives it's too early to tell whether Snowden's leaks have damaged U.S. national security. Inglis testified before the House Judiciary committee.
The Associated Press contributed to this report.