The fee is in the region of £25-30m with the deal believed to be until 2019, with the option of another year.
Subject to the 21-year-old passing his medical next week, he will join United once the transfer window opens in June.
Depay, who has scored 21 goals as PSV won this season’s Dutch title, tweeted: “Time to set my goals again on winning trophies with Manchester United.”