Manchester United, Leeds and Southampton were all linked with the winger, who was named as the Dutch Footballer of the Year in 2021/22 after scoring 21 goals in all competitions.
United were understood to be leading the race for Gakpo, but instead secured number one target Antony from Ajax in a deadline day deal.
Gakpo has began the new campaign in brilliant form and has scored eight goals and provided six assists in just seven Eredivisie fixtures so far this season.
Interest in the 23-year-old is expected to resume in the January transfer window and PSV director Marcel Brands has revealed there is an agreement to sell the Netherland international at the right price.
“I can say, with my hand on my heart, that there has been no offer for Cody to which the management or the supervisory board have said. We must do this,” Brands told Voetbal International.
“We did state that if there were certain offers, we would have to sell.
“[This is] not only from a financial point of view, but also because agreements were made with the Gakpo camp before my time.”