The Russian transfer window does not close until Monday night, allowing teams from the nation to negotiate with players despite the English alternative having slammed shut last month.
Anzhi famously sold the majority of the first-team stars – including Willian and Samuel Eto’o – last year as they set about slashing their vast spending, and have since struggled.
With the club now at the bottom end of their league, they are ready to make a desperate move to snatch a player capable of saving them from the drop.
Michu is seen as the ideal candidate, and they are ready to offer the Spaniard a lucrative deal to swap South Wales for Dagestan.
The 27-year-old has been out of action for the majority of the season with an ankle injury, but rumours indicate that he is fit enough to make a move.
Having purchased the versatile attacker for a minimal fee, the Swans may elect to cash in if Anzhi offer over £6m.
With David N’Gog and Wilfried Bony in their squad, the reliance on Michu has reduced of late.