Kommersant daily cited Putin's spokesman Dmitry Peskov as saying that Russia would consider such a request from Snowden, if it were made.
"We will take action based on what actually happens. If we receive such a request, it will be considered," he told the newspaper.
Snowden's whereabouts were shrouded in mystery as US lawmakers demanded his immediate extradition from Hong Kong over his sensational leaking of an Internet surveillance programme.
Snowden, a 29-year-old technology expert working for a private firm subcontracted to the US National Security Agency, checked out of his Hong Kong hotel after revealing his identity to the British-based Guardian newspaper on Sunday.
His explosive leaks last week of vast surveillance programs run by the NSA that trawl through telephone and Internet records have triggered widespread consternation, gaining him admirers but also critics who denounce him as a traitor.
Snowden told the Guardian he hopes to win asylum in Iceland, but the head of Iceland's Directorate of Immigration said it had received no formal request and said Snowden would have to be on Icelandic soil to make one.
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Iceland? Longtime US ally? He would get handed over to US Government in a heartbeat. He should take the Russian offer in my opinion.
He can come to Pakistan.. oh I forgot Rehman Malik is no more Interior Minister...lol