Zaha has risen to prominence this season after a string of impressive performances for Palace, while he earned a call up for England's friendly against Sweden last year.
Numerous Premier League clubs have been linked with a move for the 20-year-old, including Manchester United, Arsenal and Tottenham.
However it appears to be a straight fight between Manchester United and Arsenal for his signature, and ESPN report that it is the Old Trafford club who are closer to securing his services.
The website claims that Zaha would prefer a switch to Old Trafford over the the Emirates because of the two club's respective league positions and their relative strengths.
Despite Palace claiming earlier this season that it would take a bid in the region of £20 million to force them to sell up in the middle of their push for promotion, a price of around £10 million to make the deal happen in January has been touted.
Reports earlier this week claimed that Palace are opening up to the idea of cashing in on Zaha this month, but only if he is immediately loaned back. That is said to have harmed Arsenal's chances of landing the talented youngster as the Gunners had hoped to integrate him immediately.
The player himself says he feels ready to make the move to the next level, although Eagles co-chairman Steve Parish insists he won't be sold in January.
ESPN also report that if Manchester United do agree a deal for Zaha in January, it will spell the beginning of the end of Nani's career at Old Trafford.