The 21-year-old, who had three years left on his Palace contract, started 35 Premier League games for the club last season as they finished 12th.
Wan-Bissaka is set to be the second new arrival at Old Trafford in the close season as manager Ole Gunnar Solskjaer looks to overhaul the squad after a disappointing sixth-placed finish.
They have already signed Wales international winger Daniel James from Championship side Swansea City.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Christian Radnedge)
