The billionaire-backed Dagestan-based side are desperate to replace Christopher Samba, who signed for QPR last month, and have made the Frenchman as their number one target.
With the Russian transfer window not yet closed, Anzhi could make their move quickly in a bid to shore up their squad ahead of the new campaign.
Manager Guus Hiddink has reportedly contacted Spurs already, and is set to step up his pursuit.
Tottenham coach Andre Villas-Boas is believed to be willing to accept around £10m for the 27-year-old, with the re-emergence of Michael Dawson giving him a selection dilemma at centre-back.
Anzhi are said to be happy to meet this valuation, despite concerns over Kaboul’s proneness to injury.
The former Portsmouth man was one of Tottenham’s top performers last term under Harry Redknapp, helping the North Londoners clinch fourth spot.
But, knee surgery in August has restricted him to just one appearance this term.
Anzhi have the financial muscle to offer Kaboul an attractive contract to swap White Hart Lane for the Russian top flight, and have used similar methods to sign Samuel Eto’o, Roberto Carlos and Lassana Diarra in the past.