The 20-year-old has featured regularly in pre-season and has been named in the England squad for next week's friendly against Scotland.
Cole believes that Zaha will get his chance at Old Trafford and only needs to look at Ryan Giggs to see what he must do to be successful.
"Zaha has come in, put a good shift in pre-season and now David Moyes can make his mind up from there," he told reporters.
"Everyone starts with a clean slate so to speak, as it's a new manager with new ideas. He only needs to look at someone like Ryan Giggs and he will know what level he needs to get to be a success.
"All I would say to him is to enjoy it. It's totally different to Crystal Palace, but he's now at Manchester United and he'll have to work extremely hard to get in the team."
Zaha will hope to be part of the United squad that takes on Wigan Athletic in the Community Shield tomorrow.