According to Spanish outlet Marca, Liverpool will offer the 26-year-old a new deal before the end of the transfer window.
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The report adds that should Konate reject that deal, the Premier League side will look to sell him, with Barcelona and Real Madrid both interested.
Konate has just one year remaining on his contract at Anfield, and Liverpool are keen to avoid a repeat of what happened with Trent Alexnader-Arnold.
Liverpool are still in the market for a new centre-back regardless of what happens with Konate and are already set to sign wonderkid Leoni.