Zaha, who could make his senior England debut against Sweden on Wednesday, has been a revelation in the Championship this season and has reportedly attracted the interest of United boss Sir Alex Ferguson.
The 17-year-old has already made over 100 appearances for the Eagles but Palace are desperate to keep hold of Zaha as they push for promotion.
It has been reported that Ferguson will try and strike a deal with Palace that will allow the forward to remain at Selhurst Park until the end of the current campaign before making the move to Old Trafford.
Palace co-chairman Steve Parish has previously stated it would take a bid in the region of £20million to convince them to cash in on Zaha, who is under contract until the summer of 2017.
However, United will reportedly test the water with a bid of around £10million but they will face plenty of competition for the highly-rated forward, who has also been linked with the likes of Spurs, Arsenal, Liverpool and Manchester City.