The France international is off contract at the end of the season and has so far resisted Liverpool's attempts to discuss new terms.
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Real Madrid are watching developments and plan to offer Konate a pre-contract in January ahead of securing him in a Bosman transfer in June.
However, BILD's Christian Falk is also reporting today Bayern are also keen.
With Dayot Upamecano running down his deal at Bayern this season, the German giants see Konate as an ideal replacement for his fellow Frenchman.