Russian newspaper Izvestia has reported the cash-rich club from Dagestan, who saw the Senegal international at close hand as he dumped them out of the Europa League with an injury-time winner at St James' Park last season, are ready to offer the equivalent of £15.4 million to secure his services.
In addition, they are said to be ready to pay the player around £50,000 a week in wages and hope the lure of another season of European football, something he will not get on Tyneside this time around, will prove attractive too.
Newcastle owner Mike Ashley is not averse to selling a player if the price is right, as it was with Andy Carroll in January 2011 when Liverpool offered a staggering £35m for the England international.
However, Press Association Sport understands Anzhi would have to find far in excess of £15.4m to make the Magpies even contemplate selling Cisse, who has four years remaining on his contract.
The club has received no bids for any of its players to date and their current concentration is on incomings rather than outgoings.
Indeed, manager Alan Pardew's main objective as he headed off on his summer break was to strengthen, rather than completely overhaul, his striking department with Cisse and Shola Ameobi his only recognised senior frontmen.