The 20-year-old, who won his first England cap in November, will finish the season with Palace and link up with United in July under the terms of the agreed deal.
A statement on United's official website read: "Manchester United, Crystal Palace and Wilfried Zaha have reached an agreement for the player to join Manchester United in July 2013. The player is having a medical in Manchester today (Friday, January 25). There will be no further comment by any party until Wilfried joins United in July 2013."
Palace confirmed that Zaha had agreed personal terms and that he had been loaned back to them for the remainder of the current campaign.
United manager Sir Alex Ferguson said: "We are signing a player of potential. We always feel we develop players well. We have proved that many times. We hope the boy enjoys coming here, when he comes in the summer."
Zaha told Palace's official website www.cpfc.co.uk: "I am delighted to be joining Manchester United and ending the recent speculation. I would like to thank the owners at Palace for making the move possible, however my focus for the next five months will be solely at Crystal Palace.
"I have been at Palace for more than 10 years, I will always love the club and I want to help get them get back to the Premier League, particularly for the fans who have always been so good to me."
Palace co-chairman Steve Parish said: "I feel this is a great deal for the club and the player. 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.
"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."
Source: PA
Source: PA