According to a source familiar with the negotiations, the deal values Wan-Bissaka at £55million, a world record sum for a specialist full back.
Following a series of rejected offers from United, the agreement for the England under-21 international who completed his second season of English Premier League football last month surpasses the €57.5m Manchester City paid AS Monaco for Benjamin Mendy two summers ago.
It also ensures that United have both a right back and a left back bought from domestic rivals for record fees.
Developed by Palace's academy, Wan-Bissaka made his senior debut in February 2018, rapidly establishing himself as first-choice right back. A quick, athletic and aggressive defender, he started 38 matches last season and was credited with four assists. The 21-year-old is yet to score a goal in domestic competition.