"The asylum matter has not yet been resolved," Mr. Snowden's attorney, Anatoly Kucherena, told reporters at the airport in remarks broadcast and translated by the BBC. "You have to take our bureaucracy into account," he added.
"Any move that would allow Mr. Snowden to depart the airport would be deeply disappointing," U.S. State Department spokeswoman Jen Psaki told her daily briefing for reporters.
She said Secretary of State John F. Kerry had spoken by telephone with Russian Foreign Minister Sergei Lavrov to reinforce U.S. requests that Russia expel Mr. Snowden back to the United States, where he has been indicted on espionage charges.
Mr. Snowden reportedly has told Russian authorities he faces political charges with little prospect of a fair trial.
Earlier Wednesday, his lawyer strongly hinted to state-owned Russia Today TV that Mr. Snowden might be leaving the airport that day.
He said the 30-year-old former contract computer technician would be free to go as soon as he receives formal notification from Russian authorities that his asylum application, filed July 16, is being considered.
"I'm calling them on a daily basis. They tell me that they're about to finish the formalization," Mr. Kucherena told RT earlier Wednesday.
Once Mr. Snowden gets formal notification, he will be allowed to leave the airport transit lounge where he has been trapped since U.S. authorities revoked his passport after he fled Hong Kong on June 23, just steps ahead of a U.S. extradition request.
"There are no travel restrictions" imposed on asylum applicants, Mr. Kucherena said. "Therefore, receiving the paper will give him an opportunity to leave the transit zone of the Sheremetyevo Airport and choose a place of residence — rent a hotel or a flat. [He can] live in any place within the Russian Federation."
According to Mr. Kucherena, a decision on the application could take up to three months, and Mr. Snowden has pledged to fight any refusal in court.