Anatoly Kucherena, a prominent pro-Kremlin lawyer who participated in the fugitive's dramatic meeting with rights activists at a Moscow airport last week, said Snowden had contacted him for consultations.
"He is actively consulting with me," Kucherena told AFP, saying he last spoke to Snowden on Monday. "After the meeting we've been in frequent touch."
"We are actively consulting and I believe that he will make up his mind in the coming days," Kucherena said.
Snowden, wanted by the United States for revealing sensational details of its vast spying operations, flew into Russia from Hong Kong on June 23 and has since been marooned in the transit zone of the capital's Sheremetyevo airport.
The lawyer said he was helping the world's most famous fugitive navigate through the complexities of the Russian legislation. He was also explaining the difference between the status of refugee, political asylum and temporary asylum to Snowden.
"Before our consultations he did not have the understanding of those issues," Kucherena said. "He needs to understand what suits him and what rights and obligations a certain status will generate."
Breaking silence for the first time since he arrived, Snowden who has essentially become stateless after Washington revoked his passport, met with a handful of rights activists and lawyers at the airport on Friday.
At the meeting, he said he would file for asylum in Russia before he could work out a way to travel legally to Latin America.
Venezuela, Bolivia and Nicaragua have indicated that they would be open to offering Snowden a safe haven.
On Monday evening, President Vladimir Putin said that Snowden would leave Russia "as soon as he can" and accused Washington of essentially "trapping" the American in Moscow.
Russian officials said they had not received an asylum application from Snowden so far. A spokeswoman for the Federal Migration Service told AFP she had no information.