Off contract in June, Konate has resisted attempts from Liverpool to discuss new terms and is set to leave Anfield this summer.
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Real Madrid have also been linked with Konate, but BILD says Bayern are now in contact with Konate's representatives.
It's also suggested if Bayern can strike a deal with Konate then Liverpool will move for Crystal Palace captain Marc Guehi as his replacement.
Both Real Madrid and Bayern have pulled back from talks with Guehi in recent days.