Palace rejected the £40m bid at the weekend and are thought to be holding out for nearer £60m for the defender.
Talks are ongoing and United have indicated they will make an improved second offer.
The 21-year-old England U21 international has three years left on his current Crystal Palace contract.
Wan-Bissaka made his debut for the Eagles in February 2018 in a 1-0 defeat to Tottenham, he has gone on to play 42 Premier League matches for the south London club, who finished 12th in the Premier League last season.
The young right-back would be United’s second signing of the summer transfer window after a £15m (plus £3m add-ons) fee was agreed for Swansea’s Daniel James on June 7.