United will pay an initial £45m for the 21-year-old with a further £5m in add-ons, having made a breakthrough in negotiations with Palace.
Wan-Bissaka will sign a five-year contract at Old Trafford with the option of a further year. Pending a successful medical, an announcement is expected on Thursday.
The full-back becomes Ole Gunnar Solskjaer's second signing as United manager, following the £18m arrival of Daniel James from Swansea City earlier this month.
Solskjaer is understood to have been a great admirer of Wan-Bissaka for some time and is said to believe that youngster's style of play will suit his United.
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The Norwegian is overseeing a rebuild of United's squad this summer following the failure to achieve a top-four finish and qualify for the Champions League last year.
Wan-Bissaka was Solskjaer's priority target in defence, though United are understood to be seeking a further addition at centre-back, with Leicester City's Harry Maguire of interest.
Solskjaer also intends to add to United's midfield, following the departure of Ander Herrera on a free transfer. Bruno Fernandes, Sean Longstaff and James Maddison have all been linked with Old Trafford.
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