The Tvoi Den tabloid said that Dynamo had told Newcastle they would be willing to let the 27-year-old Russian international go for around $9 million, half the player's asking price for Russian clubs.
Newcastle, in or around the relegation zone all season, are currently one place and three points away from safety with four games of the season remaining.
Kolodin impressed at last summer's Euro 2008, his long-range thunderbolt shots troubling Dutch keeper Edwin van der Saar during Russia's 3-1 quarterfinal win over Holland.
Russian players are currently enjoying their highest profile since the break-up of the Soviet Union, with Andrei Arshavin having made a major impact at Arsenal and Roman Pavlyuchenko turning in occasional match-winning performances for Tottenham Hotspur.
