The winger is open to moving back to Old Trafford, where he previously spent 18 months of his career but made just two Premier League appearances.
Palace are reportedly demanding a minimum of £100m for Zaha, a sum which would represent an incredible £94m profit.
The south Londoners had a solid Premier League season under Roy Hodgson, finishing 12th and never looking in danger of relegation.
Zaha was the club’s undisputed star performer, scoring 10 goals and supplying five assists for his team-mates.
And the 26-year-old has now informed Eagles chiefs that he wants to leave the club in pursuit of Champions League football.
Palace are reluctant to sell their star however, and they hope to deter Manchester United by slapping the £100m price tag on him, The Telegraph say.
Despite failing to make any impact at the Theatre of Dreams between 2013 and 2015, Zaha believes he is now ready for the challenge.
He would be competing with the likes of Anthony Martial, Jesse Lingard and Marcus Rashford for a place Ole Gunnar Solskjaer’s side.
But the Ivory coast international is supremely confident in his own ability, and said in April that he believes he has more to offer at a bigger club.
“I have to experience the Champions League,” he told the Daily Mail. “I just need the opportunity, that’s it. And I’ll do the rest.
“I’m too ambitious. And not ambitious just to play for a top club, but to win things at club level and with the Ivory Coast.
“For me to be better, to achieve what I know I am capable of, I have to aim to play at the very highest level, to win trophies.
“I’m blessed to have come this far in my playing career. But I feel like there is so much more I have to offer.”
Both Arsenal and Borussia Dortmund had previously declared their interest in Zaha, but have now admitted defeat.
The Gunners are expected to move instead for Bournemouth winger Ryan Fraser, who would cost significantly less than the Palace man.
United are monitoring Zaha’s situation, the Telegraph add, but it is unclear whether they will move for him as a deal for Swansea’s Daniel James is being pursued, although hopes of landing Borussia Dortmund’s Jadon Sancho appear over for this summer.