BBC Sport understands talks between the two clubs are continuing although, as yet, no deal has been agreed.
United manager Ole Gunnar Solskjaer is keen to bolster his squad for the new season.
He has identified 21-year-old Wan-Bissaka, who is currently on England Under-21 duty in Italy, as having the youth and speed he is looking for.
It has been suggested Palace want around £50m for the Croydon-born defender, who joined the Palace academy when he was 11 and made his first team debut in 2018.
Solskjaer said earlier this summer that he wanted his transfer business completed early in the window but, with his players due to report for training on 1 July, so far, the only new arrival has been Wales winger Daniel James, 21, who was signed from Swansea earlier this month for £15m.