Gakpo's representatives have been alerted to United's interest by recruitment staff at Old Trafford but the club have yet to submit a bid.
Gakpo is valued at around £35 million by PSV, who are managed by former United striker Ruud van Nistelrooy, and sources have told ESPN they are unlikely to accept anything less for one of their star players.
The 23-year-old came through the club's academy and scored 21 goals in 47 games last season. He has been capped seven times by the Netherlands, scoring twice.
Sources added there is interest from other Premier League clubs in Gakpo, who scored twice in PSV's 4-1 opening-day win against FC Emmen on Saturday, but that he would be keen to move to United if terms are agreed.
United have turned their attention to Gakpo, who impressed manager Erik ten Hag while in charge at Ajax, after being priced out of a move for Ajax winger Antony and missing out on RB Salzburg forward Benjamin Sesko, who will join RB Leipzig next summer after his signing was announced on Tuesday.
A fee of £15.5m rising to £19.5m has been provisionally agreed with the Serie A club and negotiations on personal terms are underway.
Sources added that Ten Hag called Rabiot to speak with him on Monday and the Juventus midfielder appreciated the conversation with the Dutchman.
Centre-back Eric Bailly is hoping to make progress on his desire to leave Old Trafford later this week, sources added.
Bailly was left in the stands for the 2-1 defeat against Brighton on Sunday and he is keen to seal a move away before the transfer window closes.
ESPN's Julien Laurens contributed to this story.