The Dutchman is a top target for the Red Devils, who will be looking to recruit a new attacker in the transfer market following the departure of Cristiano Ronaldo.
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Gakpo could be likely to leave PSV in the coming months, although director of football Marcel Brands has now issued a warning to any sides who may be interested in his signature.
The Eredivisie outfit's current record fee for a player is around £39million when Hirving Lozano moved to Napoli in 2019, but a move for Gakpo is set to fetch a much higher figure.
Speaking on PSV supporter chat show 'Eeuwig Trouw' via Football365, Brands revealed: "You have to be realistic. When you talk about Cody Gakpo and the price tag, how many clubs are able to buy a player of that calibre?
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"You are talking about maybe 10 or 12 clubs in Europe. You have to need a player in that position and be able and willing to spend the money in January, which is normally less hectic than the summer. We have yet to see if it all happens. He has no clause or anything.
"And no amounts have been agreed on either. The agreements that were made were there before my time. Amounts have not been discussed, but if the right club comes that Cody would like to go.
"Those agreements were already made before I came. It must be a record transfer for PSV anyway. Then you know in which direction it should go."
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United were linked with a move for Gakpo during the summer, but Erik ten Hag opted to raid former club Ajax to sign Antony instead.
It's clear that their interest in the winger has by no means diminished, with the PSV forward having a superb season so far.
He has 13 goals and 17 assists in 24 games across all competitions for the club, making him one of European football's hottest prospects.
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Gakpo also had a superb World Cup campaign with the Netherlands, scoring three goals in the group stages before they were eventually knocked out by Argentina in the quarter-finals.
The player himself has not ruled out a move in the January transfer window, saying that he wants to 'reach the highest level'.
United already spent over £200million in the summer on new signings but should they wish to also bring in Gakpo, then their expenditure for the season could rise even higher.