"He has reached his location. He feels fine and is completely safe," lawyer Anatoly Kucherena said.
On Thursday, Snowden was granted a one-year permit to stay in Russia following a month-long confinement in the airport's transit area.
An official invitation will also be sent to Edward's father Lon Snowden to come to Moscow, Kucherena said Friday.
"We have prepared almost all the documents. Invitations are to be sent to Lon Snowden and his defence attorney on my behalf to come to Moscow," Kucherena said.
Kucherena said that due to security reasons the time "has not come yet" to disclose Snowden's location.
Also on Friday, US Ambassador to Russia Michael McFaul met presidential aide Yuri Ushakov to discuss the latest developments in bilateral relations and the status of Snowden, the US diplomatic mission said.
"Michael McFaul and Yuri Ushakov discussed nuclear arms cuts, anti-missile defense, Syria, trade, human rights and the new status of Mr. Snowden," the US Embassy said on its Twitter account.
The White House had Thursday expressed "extreme disappointment" at Russia's decision, saying Washington was "evaluating" a scheduled presidential summit planned for September.