According to Foot Mercato, the 21-year-old French youth international has been convinced by new manager Vincenzo Italiano, who played a key role in the negotiations.
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While talks between the two clubs are still ongoing, a transfer fee of around €10 million is expected to be enough to complete the deal.
Monaco, however, remain open to considering other offers, with a Premier League club also showing interest, although their pursuit is said to be less advanced than Besiktas's.
Ouattara, who is under contract until 2028, made 24 appearances across all competitions last season, including seven in the UEFA Champions League.