Gaitan has been attracting interest from a number of top European clubs following his impressive displays for Benfica.
The Argentine international has been heavily linked with a move to the Premier League with both Liverpool and United keeping a close eye on his progress.
But reports in the Russian media claim Anzhi are now stepping up their efforts to sign the 24-year-old and would be ready to pay over £20m for his signature.
Benfica are determined to keep their star midfielder at the club but will not stand in his way if a sizeable offer is made when the transfer window re-opens in January.
Gaitan's agent Jose Ibarren said: "I do not rule out the possibility of him leaving Benfica if there is a big offer."