The 23-year-old Netherlands international is the latest name to be linked with Old Trafford as new manager Erik ten Hag seeks to strengthen his squad in the final weeks of the summer transfer window.
United are yet to open talks with Gakpo or PSV but - as reported by The Independent on Tuesday - their interest in the winger is known.
Gakpo played as PSV's progressed in the Champions League qualifying rounds with a 3-2 win over Monaco on Tuesday evening. They now face Rangers for a place in the group stages.
"We must first focus on and try to make it to the Champions League," Gakpo told ESPN after the game when asked about his future.
"Then it is more likely that I will stay. I don't think I have said anywhere that I am leaving, so there is definitely a chance that I will stay.
"I'm open for a cup of coffee with the management - or coffee for them and water for me. We'll see how it goes."
Gakpo is a right-footed winger who has predominantly played on the left for PSV, who are managed by former United striker Ruud van Nistelrooy.
If United choose to firm up their interest, Gakpo would represent yet another Eredivisie-based signing under Ten Hag, following the arrivals of Tyrell Malacia and Lisandro Martinez from Feyenoord and Ajax respectively.
Christian Eriksen, United's other summer arrival, is a former Ajax player who also trained with the club last season while recovering from heart surgery.
United ended their pursuit of Marko Arnautovic on Monday following a fan backlash over the potential signing of the Bologna forward.
Arnautovic had emerged as a surprise target amid uncertainty over Cristiano Ronaldo’s future at Old Trafford and a lack of options up front following a hamstring injury to Anthony Martial.