United and Tottenham are both tracking the 20-year-old winger, who impressed under Van Gaal at the World Cup as he scored two goals in Netherlands' run to the semifinals in Brazil.
MORE: Beautiful fans in Brazil | Barcelona's new bright kits | Juve's jerseys
PSV has tried to ward off interest in Depay by setting a 22.5 million euro asking price for the player and the club is keen to keep him at the club.
Spurs have shown the strongest interest as new manager Mauricio Pochettino looks to improve his attacking options and Depay is keen to test himself in the Premier League.
But Van Gaal is a huge admirer of Depay, having worked with him at close quarters with Oranje and selected him for the World Cup squad.
The new United boss has told Depay in a private conversation that his career would be best served by remaining at PSV for another season before seeking a transfer.
United is unlikely to pursue a deal for Depay this summer but will continue to monitor his situation with a view to a potential deal next year.
Real Madrid has also been linked with the Dutchman, who scored 12 goals in 32 league appearances for PSV last season.