The 20-year-old will try to fight for a place in David Moyes's team following his arrival from Crystal Palace this summer.
"I think we're all looking forward to seeing Zaha," Cole told the club's official website.
"He's a young, up-and-coming player and he did extremely well at Crystal Palace last season and most people I've spoken to are looking forward to him playing and being involved this year. Of course, there's a danger you can expect too much of him."
Zaha signed for the Red Devils in January but remained at Selhurst Park for the remainder of the campaign, helping the Eagles reach the Premier League by winning the Championship playoff final.