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United handed summer transfer funds for Red Devils boss Ole Gunnar Solskjaer to land Jadon Sancho, Raphael Varane and Cristiano Ronaldo.
However, United have faced growing criticism over their failure to sign a midfielder ahead of the current Premier League season.
The Red Devils have been criticised for the quality of their holding midfielders, with Scott McTominay, Fred and Nemanja Matic failing to impress on a weekly basis.
Solskjaer also faces the possibility of losing both Pogba and Jesse Lingard, whose respective contracts will expire at the end of the season.
Pogba, 28, has been linked with Juventus, Real Madrid and Paris Saint-Germain, with United pushing for the Frenchman to sign a new deal.
Mundo Deportivo now reports that United have identified Kessie as a potential target, having previously been linked with the Milan star.
Kessie was initially loaned to Milan from Atalanta in 2017 and has since become an important player for the Serie A side.
The 24-year-old Ivorian midfielder, who made the transfer permanent in 2019, was valued at around £40m when he was with Atalanta.
But Kessie’s current deal with Milan will expire in June 2022, with his long-term future at the San Siro plunged into doubt.
According to Mundo Deportivo, several clubs have been put on red alert and are prepared to make a move for the Ivory Coast international should he leave Milan.
Kessie is free to speak with foreign clubs about a potential summer transfer from January, with United closely monitoring his situation.
The Milan midfielder has made 187 appearances for his current club and scored 30 times and registered 15 assists.
United have also been linked with Leicester City midfielder Youri Tielemans and West Ham star Declan Rice.