Snowden's lawyer Anatoly Kucherena declined to give the name of the company but attention centred on VKontakte which is seen as the Russian equivalent of Facebook.
"Edward Snowden will start working at a big Russian company on Friday, November 1. His job will be to support and develop a major Russian website," Kucherena told the Interfax news agency.
(Also see: Edward Snowden gets job offer from Russia's top social networking site)
The charismatic founder of VKontakte Pavel Durov, often described as the Russian Mark Zuckerberg after the equally youthful Facebook supremo, publicly offered Snowden a post in August.
"We invite Edward Snowden to Saint Petersburg and will be glad if he decides to join the star team of programmers," Durov wrote on his social networking page on the same day as Snowden received asylum in the country.
Twenty-nine-year-old Durov, who co-founded the social networking service in 2006, suggested Snowden could work on "protecting the personal data of millions of our users".
VKontakte, or In Touch, says it has over 100 million users.
VKontakte spokesman Georgy Lobushkin told RIA Novosti news agency he could not comment on speculation that Snowden would work there.
Two other major Russian Internet companies, Mail.Ru Group and Yandex, issued outright denials, saying they had not hired Snowden, RIA Novosti reported.
Snowden spent more than a month in a Moscow airport before receiving a year's temporary asylum August after exposing massive surveillance by the US National Security Agency.
His lawyer has said in interviews that the fugitive is running short of money. So far a supporters' website has raised almost $49,000 in donations.
Snowden willing to 'shed light' on US spying in Germany
Lawyer Kucherena told Interfax that her client would not respond to questions from foreign secret services about US spying which has caused an uproar after revelations of eavesdropping on foreign leaders.
"According to an agreement, while he is in Russia, he cannot reveal secret information," she said, responding to a question on the possibility her client could be questioned over suspected eavesdropping on German Chancellor Angela Merkel
However German public television ARD reported Thursday that Snowden had met with Green Party lawmaker Christian Strobele in a secret location with two journalists.
"He proved that he knows many things. He is willing in principle to help shed light" on the spying, Strobele told the station.
He added that Snowden declared his willingness "to come to Germany to testify" if the right conditions were in place, or to testify in Russia.
The most recent revelations from intelligence leaked by Snowden came in a Washington Post report alleging that NSA technicians had tapped into Yahoo and Google data centres around the world, winning access to vast amounts of private data.
Wanted for espionage by Washington, Snowden has been living in hiding and keeping off the radar in Moscow with what his lawyer called "small alterations in his appearance."
Popular Russian website Life News published a photograph on Thursday apparently showing Snowden on a boat trip in central Moscow, without glasses and wearing a cloth cap.
The website, which specialises in sensational scoops, said the photograph was taken in September and that it paid 100,000 rubles ($3,122) for the image.
The same website earlier this month published a photograph of Snowden pushing a shopping trolley, later confirmed as authentic by Kucherena.
Life News, which has close contacts with law enforcement officials, said the photograph "proves that the former US agent either lives permanently in Moscow or visits regularly".
A video was released this month showing Snowden meeting four US whistleblowers, who presented him with an award in an unidentified location that appeared to be in Russia.
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