(CNN) -- Edward Snowden isn't yet allowed to step outside the Moscow airport where he's been holed up for weeks, despite reports to the contrary, his Russian lawyer, Anatoly Kucherena, said Wednesday.
Russian media had reported Wednesday that the U.S. intelligence leaker was issued a document that would allow him to wait elsewhere in Russia while his request for temporary asylum was considered.
But Kucherena, after meeting with Snowden in Sheremetyevo International Airport's transit area Wednesday, told reporters that Snowden hadn't received the certificate and that he would remain in the transit area for now.
That certificate still could come at "any time," Kucherena told CNN.
The news is the latest development in Snowden's search for a place to settle after the United States charged him with espionage.
The former National Security Agency contractor, who admitted last month to revealing sweeping U.S. electronic surveillance programs to the news media, left Hong Kong for Moscow on June 23. Since then, he's been unable to leave the airport's transit area because the United States revoked his passport.
White House spokesman Jay Carney said Wednesday the U.S. government is seeking "clarity" about Snowden's status. And a spokeswoman for Secretary of State John Kerry said Washington would find it "disappointing" if Snowden were allowed to leave the airport.
Kerry spoke with his Russian counterpart, Foreign Minister Sergey Lavrov, on Wednesday morning, spokeswoman Jen Psaki said.
"He reiterated our belief, which we stated publicly and privately, that Mr. Snowden needs to be returned to the U.S., where he will face a fair trial," Psaki said.
Snowden applied for temporary asylum in Russia on July 16. If granted, he would be able to live in Russia, and even travel abroad, for at least a year, Kucherena said last week.
A ruling on the application could take months. But Kucherena has said that the Russian government could issue him a certificate that would allow him to leave the airport and wait somewhere else in the country while the application is considered.
On Wednesday, Kucherena said that he is in daily contact with Russian authorities about securing Snowden permission to leave the airport, state-run media outlet RIA Novosti reported.
If Snowden is granted temporary asylum in Russia, it's unclear whether he'd try to move elsewhere. He's previously indicated that he eventually wanted refuge in Latin America. But Kucherena suggested last week that Snowden might take his time in Russia.
"As far as I know, he's planning to stay in Russia to learn Russian culture, Russian language and (to) live here," Kucherena told CNN last week.
Washington has no extradition agreement with Russia, and FBI agents who work at the U.S. Embassy there have no authority to make arrests.
Russian President Vladimir Putin said this month that Snowden would need to "stop his work aimed at harming our American partners" if he wanted to stay in the country.
In a subsequent meeting with human rights activists and lawyers at the airport on July 12, Snowden reportedly said he wanted temporary asylum in Russia while awaiting safe transit to Latin America, and added that he would not harm the United States in the future.
The presidents of Venezuela and Bolivia have said their countries would give Snowden asylum, and Nicaragua's president said he would offer it "if circumstances permit." But he would need the legal ability to travel there -- something that temporary asylum in Russia could give him.
The U.S. government has asked Russia to expel Snowden. Absent that, it will watch carefully the route he takes if he tries to reach one of the Latin American countries willing to take him in.
The United States could grab Snowden if any plane carrying him were to refuel in a country that respects U.S. arrest warrants. But he probably will be careful to avoid that scenario.
Nevertheless, the United States has sent provisional arrest warrants to a number of countries where Snowden could either transit or seek asylum, a U.S. official said last week.
CNN's Phil Black, Alla Eshchenko, Jason Hanna and Carol Cratty contributed to this report.