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.
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.
