United have been given permission to speak with the England international by Crystal Palace and Press Association Sport understands he will travel north with his advisor Will Salthouse to conclude the deal.
It is anticipated the 20-year-old will immediately be loaned back to Palace for the remainder of the season.
Sir Alex Ferguson is a long-time admirer of Zaha and last week watched him in Palace's FA Cup third round replay defeat at Stoke.
In addition, Ferguson's assistant Mike Phelan was at Burnley for a recent Championship game, and with Arsenal manager Arsene Wenger confirming earlier this week that the Gunners were not interested in the young wide-man, United had the way clear to seal a deal.
There was a brief delay as the Red Devils were unable to conclude the transfer in the same manner in which they secured the services of Chris Smalling from Fulham two years ago, when they agreed the deal but left him to finish the season at Craven Cottage before completing the transfer.
However, those issues have now been resolved and the deal is expected to be finalised before the weekend.