Brazilian-born Douglas - who is now set to represent Holland - was previously close to a deal with Newcastle.
But now Anzhi have made a move for the 25-year-old as they look to replace Samba, who quit the club last month to move to Queens Park Rangers. The Russian transfer window is still open until February 27.
Douglas is out of contract in the summer, so this would represent Twente's final chance to get money for the player given he can leave for nothing at the end of the season.
"Anzhi have made a serious offer for my client," Douglas's agent Rodrigo da Silveria confirmed.
"I think we'll need about a week to make the decision about Douglas' future.
"He doesn't want to leave Holland but he wants new challenges. Anzhi are a big club with world-class players. It's possible that in February Douglas will be part of the Russian team."