PSV Eindhoven do not want to let Gakpo depart, but failing to reach the group stage of the Champions League could see his departure.
Manchester United view Gakpo as an alternative to top target Antony, while Everton have scouted him and may need to replace Anthony Gordon before the transfer window shuts.
PSV Eindhoven want to keep him however and, according to Dutch daily the Eindhovens Dagblad, a new five-year contract is waiting.
The five-year deal would make Gakpo one of the highest paid players in the Netherlands.
This week is set to be key, not least because PSV Eindhoven will learn if they have reached the group stage of the Champions League.
They are in action against Rangers at Ibrox, with the first leg in the Netherlands having ended in a thrilling 2-2 draw.
Gakpo’s agents have already agreed on the deal with PSV Eindhoven, but this week will see if the player signs it.