Off contract in June, Konate is being linked with Europe's biggest clubs, including Bayern, Real Madrid and PSG.
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BILD's Christian Falk is reporting Liverpool are ready to sell Konate in the New Year if he continues to refuse to agree terms over a new contract.
He states: "Bayern will be watching very closely how the decision is made. Liverpool are putting on the pressure - they definitely want to sell him if he doesn't sign a new contract soon.
"They already have one on the table. They absolutely don't want to lose a player on a free transfer. They would then immediately invest the money in Marc Guehi from Crystal Palace."