The French defender has recently returned from injury - he hasn't featured for Spurs since the opening game of the season after requiring surgery on a knee complaint.
But Anzhi are keen to secure a replacement for the departed Christopher Samba - the former-Blackburn centre-back joined Queens Park Rangers in the January transfer window.
The Guardian suggests Kaboul would stand to earn considerably more than his current £35,000-a-week salary should he join the big-spending Russia club.
But Spurs have rejected Anzhi's advances, as they feel the Frenchman has a role to play as Tottenham look to seal a top four spot and a place in next year's Champions League.
The Russian transfer window closes at midnight on Wednesday night, so Anzhi face a race against time to shore up their defensive options following the sale of Samba.
The France international's contract expires at the end of next season, and the report claims he's likely to be offered an extension by Spurs in the summer - provided he can prove he's returned to full fitness.
The Russian league season resumes on 10 March after a lengthy winter break, and Anzhi have been one of the most active clubs in the transfer window - signing Shakhtar Donetsk star Willian and Lokomotiv Moscow fullback Andrey Yeschenko.