Konate is inside the final 12 months of his contract and refusing to consider new terms offered by the Reds.
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Real Madrid are keen on the France international, who is also eager to make the move, says AS.
Real Madrid are willing to offer €20-25m to sign Konate before the summer market shuts. However, Liverpool rate the defender closer to €50m.
Indeed, Reds management haven't given up on securing the Frenchman to a new deal and will try again before considering any offer from Real.