Konate, 26, recently rubbished talks he was set to put pen to paper on a new contract at Liverpool, with his current deal set to expire at the end of the season.
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Real Madrid have been heavily linked with a move for the Frenchman, hoping to pull of a similar deal that saw them sign Trent Alexander-Arnold this summer.
The centre-back is currently away on international duty and revealed that superstar Mbappe is pulling out all the stops to persuade him to join him in Madrid.
Speaking to Telefoot, Konate said Mbappe "calls me every two hours to sign for Real Madrid."