Zaha was widely expected to join Cardiff after undergoing a medical at the weekend and he posted a photo of himself in the club's training kit via social media.
Neither team have confirmed the deal, but he trained with Cardiff on Thursday and the completion of the move is thought to be a formality.
The 21-year-old made a £15m move to Manchester United last January before being loaned back to Crystal Palace and helped the Eagles secure promotion through the play-offs.
But Zaha has failed to establish himself in United's side this season and is yet to make a Premier League start for the club.
The England international will link-up with Manchester United team-mate Fabio, who joins Cardiff on a permanent basis for an undisclosed fee.