United and PSV have confirmed the transfer and Depay, 21, will move to Old Trafford - pending a medical - when the transfer window opens in June.
United said via the club's official Twitter page: 'Manchester United has reached an agreement with PSV Eindhoven & Memphis Depay for the player's transfer, subject to a medical.
'Pending the successful completion of the medical, the deal will go through 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.
PSV boss Philip Cocu announced the move to his squad on Thursday morning.
Sporting director Marcel Brands told PSV's official website: 'We are very proud of this upcoming transfer. Memphis has gone through all the teams of the PSV youth academy and has played an important part in us winning the league title this season.
'He's a wonderful product of our club. He'll make a good transition to a magnificent club in a beautiful competition. We obviously lose a fantastic player, who is of great value to the team but we are very happy to grant him this move.'
Despite reports that the player and his entourage had already visited United's Carrington training complex for meetings with club officials, it is understood no meetings had taken place.
United posted their interest formally last month when executive vice chairman Ed Woodward telephoned PSV Eindhoven.
source:dailymail