The forward arrived at the club from Crystal Palace last January, spending the remainder of the season in the Championship as he helped push the club to victory in their play-off final triumph against Watford.
The 20-year-old has already earned an international cap with England but Cole – who was part of the famous treble-winning side of 1998-99 – has warned fans not to put too much pressure on their new man to perform.
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"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."
The 20-year-old has been a key feature during the club's pre-season tour this summer and fellow treble winner Dwight Yorke believes that Zaha is suited to the club as he plays "the United way".
"To be a United player, the club will have done a lot of research on him," Yorke remarked.
"He's coming into a new setup and new surroundings. On tour, when he came on, he lit up the stadium because he just loves to attack. That's the United way."