Zaha was Sir Alex Ferguson's last-ever signing in January but spent the remainder of the campaign at Crystal Palace.
After helping the Eagles reach the top-flight and representing England at the Under-21 European Championship, the winger joined up with his new team-mates this week.
"It's down to the new manager," explained Cole when asked about the recent recruit at the press conference announcing Aeroflot's new partnership with United.
"Obviously, Sir Alex bought him last season and it's up to Zaha to come here and put in a shift in pre-season, give it a good go and David Moyes will make his mind up from there.
"I think it's a good thing that everyone is starting from a clean slate, so to speak, as the new manager is coming in with new ideas and he'll pick the team as he feels fit for going to win the championship week in and week out.
"I think it's a great opportunity for Wilfried and that's why he's come to Manchester United in the first place - to have that opportunity - and no doubt he'll enjoy it."