The Ivory Coast-born 20-year-old, who left west Africa as a four-year-old, will be immediately loaned back to Crystal Palace for the remainder of this season.
Palace said they expected the deal to be concluded early next week and that Zaha had agreed personal terms with the 19-times English champions.
"We would like to thank Wilf for all he has done for the club and we all wish him well when he joins Manchester United in the summer," Palace co-chairman Steve Parish said on his club's website (www.cpfc.co.uk).
"It was essential to Wilf and ourselves that he stayed at Palace for the remainder of this season to help the club with its main aim of promotion to the Premier League."
United, who confirmed the deal on their website (www.manutd.com), said the player was having a medical in Manchester on Friday but that there would be no further comment until he joined the club in July.
Neither club gave any financial details but British media have widely reported the deal to be worth 15 million pounds ($23.68 million).
Zaha becomes United's first signing of the January transfer window.
His displays for Palace have caught the eye of the Premier League elite, with Arsenal also thought to be in the running to seal a deal until manager Arsene Wenger this week said he was not interested.
Zaha made his England debut as a substitute in the 4-2 friendly international defeat by Sweden in November last year, becoming only the second Championship (second division) outfield player in five years to be called up by England.
He is still eligible to play for the Ivory Coast because the game in Stockholm was classed as a non-competitive game by FIFA.
($1 = 0.6336 British pounds)
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