The deal is worth £50 million and the 21-year-old England Under-21 international is set to travel for a medical imminently before going on holiday, BBC Sport reported.
United has offered Wan-Bissaka a long-term contract and wages of up to £80,000 a week.
He is on £10,000 a week at Selhurst Park, the lowest-paid player in Palace's first-team squad.
Wan-Bissaka joined Crystal Palace academy when he was 11 and made his first-team debut in 2018.
Manchester United manager Ole Gunnar Solskjaer has been keen to bolster his squad for the new season and identified Wan-Bissaka as having the youth and speed he is looking for.
The defender had been on England duty at European Under-21 Championship but, after the Young Lions were knocked out at the group stage, a deal seems set to be swiftly concluded.
If the deal is completed, Wan-Bissaka will be United's second summer signing, after 21-year-old Wales winger Daniel James joined from Swansea for £15 million earlier in June.