It is understood that the Serie A side are looking to offload the Guinean ace, who has been a bit part player at the San Siro.
Although he is a midfielder by trade, the 26-year-old has often filled in at left-back for Milan, and is known for his ability to operate well in a number of roles.
With his club willing to sell for around £3m, Tim Sherwood is tempted to launch a move in the next transfer window.
The Spurs chief is keen to bolster his squad at the next opportunity, and sees Constant as a potentially valuable addition.
A number of other clubs are also interested, including Everton and Borussia Dortmund.
Constant has been lauded for his athleticism and power, which could be two vital characteristics in the Premier League.
Tottenham’s hopes of landing the payer may be dependent on their ability to finish in the top four, with the player likely to be keen on playing Champions League football.