The 20-year-old is the Championship's hottest property, with Chelsea, Arsenal, Liverpool and Tottenham also reported suitors.
United have yet to make an official bid for the player, who made his England debut against Sweden in November, but it is understood Sir Alex Ferguson has been monitoring Zaha since he dazzled in Palace's shock League Cup win at Old Trafford in 2011.
Zaha has already stated his desire to play in the top flight, having made more than 100 Championship appearances.
"I'm not a teenager any more," he said. "I've played more than 100 first-team games and this year is crucial to develop my football career. I want to establish myself in the Premier League and in international football.
"I've played a lot in the Championship so having that new challenge in the Premier League, where everyone wants to be, would be great.
"To be honest, if I had the chance in January and Palace agreed, it's something I'd want to go for."