The France defender's current deal is due to expire at the end of the season.
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He has been linked with the likes of Real Madrid and Bayern Munich, but the Athletic says Konate is now warming towards committing to a new Reds contract.
There has not yet been any breakthrough in negotiations between Liverpool and Konaté.
The situation therefore remains uncertain and there is still some way to go before the parties can find agreement on a new agreement.
Konate is demanding a signficant rise on his current £150,000-a-week deal to re-sign with the Reds.
