United have been closely linked with the Ivory Coast international since this summer due to their lack of strength in midfield, and with Paul Pogba set to depart Old Trafford at the end of the season.
However, the Red Devils may be priced out of securing Kessie’s signature.
According to ESPN, Spurs are set to offer the midfielder a lucrative wage in a bid to persuade him to leave Italy for North London.
Kessie has been offered a £5.4million-per-year contract by Milan to try and keep their star at the San Siro, but it is believed that Daniel Levy is willing to better that.
Manager Antonio Conte is expected to be given a significant budget to improve his squad over the next 12 months, and the arrival of Kessie may not make a major dent in the pot given that there will be no transfer fee to consider.
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The signing of Kessie, if he does accept the offer, could be at the expense of midfielder Tanguy Ndombele.
The Frenchman has struggled for minutes under Conte so far this season and has since been linked with a move away from the Premier League.
Speaking ahead of his side’s match against Watford this weekend, Conte was asked about Ndombele’s situation, to which he bluntly replied: “He’s a midfielder.”
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Then pressed for more details, the Italian continued: “I pay the same attention in every single session to every player. My concentration is not only on 12, 13, 14 players but the full squad. There are players that understand very well, others need more time or for others, it could be difficult to understand.”
Conte was then asked about whether the former Lyon star may leave Spurs, to which he replied: “I think it’s better for me and the club to speak together and then we’ll have the situation more clear than now. To speak about a player that can come or go, now it’s not right, it’s not honest.”
It remains to be seen whether Kessie buys into Conte’s vision and joins Tottenham, or whether the midfielder waits for more other offers from elsewhere before deciding where his future lies.