The Red Devils are very eager to add a defensive minded central midfielder, either in January or in the summer transfer window.
His current club Milan have offered Kessie a new contract, but the midfielder is not happy with their offer.
Kessie seems unlikely to have any desire to renew with the Italian giants, despite their improved form in Italy and Europe over the past 12 months.
The Ivory Coast star's agent George Atangana told Calciomercato.com: "An important player needs a contract that is up to his standards.
"There was a formal offer from the club and a request from the agent, but we are talking about figures a long way off what you are referring to [£6.5million a year].
"For too long people have been saying things in the wrong way. I am sorry for those who read these stories and believe them to be true."
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