The Turkish club are set to splash the cash this summer for new manager Jose Mourinho.
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The ex-Chelsea, Manchester United and Tottenham coach wants to sign Lerma for a reasonable fee.
He is also, per Sporx, set to target Tottenham duo Pierre-Emile Hojbjerg and Rodrigo Bentancur.
Fenerbahce would be willing to pay €15 million for Lerma, who has a contract until 2026.
Whether Palace agree to sell is unclear, as they have already lost Michael Olise and may sell Eberechi Eze.