The France international is now inside the final 12 months of his current deal and is said to be planning to run down his contract.
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Konate is a Bosman target for Real Madrid and has also talked up the prospect of returning to France at PSG.
The Mirror says Konate has fielded an offer from Saudi Arabia, but is insisting he wants to remain in Europe at this stage in his career.
Liverpool insiders are becoming increasingly resigned to Konate playing out the final 12 months of his deal.
