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Kessie is regarded as one of Serie A's top midfielders and he's free to negotiate with any overseas club from today.
The 25-year-old is being monitored by a number of top teams across Europe and Italian outlet ANSA claim Jurgen Klopp's side are already working on a deal.
Kessie has made 16 appearances in the league this season and has scored five goals. He also played against Liverpool in the Champions League twice this season.
Milan are desperate to see the Ivory Coast international stay but have accepted he'll likely be moving for free in the summer.
The club are currently four points behind Inter in Serie A but crashed out of the Champions League after finishing bottom of Group B.
Liverpool have had issues in midfield this season due to injuries and positive Covid tests.
Teenager Tyler Morton has been given opportunities with Thiago, Curtis Jones and Naby Keita having spells on the sidelines.
Klopp knows he needs more quality depth if he wants to challenge Manchester City at the top of the Premier League.
This week saw Liverpool slip to a 1-0 defeat against Leicester City and it leaves them trailing the leaders by nine points going into the new year.
Liverpool play second place Chelsea on Sunday and Klopp believes whoever wins that game will be in the best position to stop City retaining their title.
"City are flying," he admitted to Sky Sports.
"Before the Leicester game, we were in a perfect position, only three points behind City, which, with the season they've played, is incredible.
"Our start to this year, 2021, was massively difficult, and to find a way back into your normal performances and maybe improve them was a massive improvement.
"But one of us can put pressure on City, the winner can… the loser I don't know.
"We will see that later on.
"We will not give up just because we lost our game against Leicester. We will try everything, 100 per cent. There are a lot of games to come."
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