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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."
He added: "I always say; dream, believe, achieve. I was dreaming and believing of becoming the champions of the Eredivisie with PSV and we achieved it.
"I'm extremely grateful towards PSV and all the fans for believing in me."
Depay, who played under United manager Louis van Gaal during his time as Netherlands boss, had attracted interest from a number of clubs across Europe, with Paris St-Germain and Liverpool reportedly keen to sign him.
Tottenham reportedly had a bid for the player turned down last year.
He has spent his entire career to date with PSV, for whom he has scored 42 goals in 104 appearances.
"We are very proud of this upcoming transfer, which came in a huge momentum," PSV's sporting director Marcel Brands told the Dutch club's official website.
"Memphis has gone through all the teams of the PSV youth academy and has played an important part in the national title this year.
"He's a wonderful exponent of our training and will make a nice transition to a magnificent club in a beautiful competition.
"We obviously lose a fantastic player, which is of great value to the team. But PSV grants him this wonderful step."
Depay played at the 2014 World Cup in Brazil as the Netherlands finished third before Van Gaal quit as head coach to take over at Manchester United.
He made his international debut on 15 October 2013, coming on in the last minute of the Netherlands' 2-0 World Cup qualifying win in Turkey.
Manchester United have agreed a deal to sign Dutch forward Memphis Depay from PSV Eindhoven.