Zaha was signed by United in January but was loaned back to the Eagles until the end of the season, which finished in promotion after a play-off final victory in which he won the match-winning penalty in extra time.
The 20-year-old won the PFA Championship Player of the Year award having finished the season with eight league goals, and made his debut for the England national team in November, but insists his sole aim now is simply to earn the right to play at Old Trafford.
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"I just want to be able to play regularly. If I manage to accomplish that then I can move on to other things. I’m just taking things step by step."
Zaha is now on tour with the Red Devils and struck the post as a substitute against an all-star XI in Thailand before starting United's next match against an all-star XI in Australia.
He admits he is still surprised to appear alongside players he has previously watched on TV, but insists he is focused on improving and hopes to score a goal in pre-season.
"I already think it’s a great achievement for me to be able to wear the Manchester United shirt. I’m just taking everything slowly and I hope to score a goal on the tour. That would cap everything for me," he continued.
"In the changing room I'm sitting between Ryan Giggs and Danny Welbeck, famous people who I watch all the time on TV.
"It's a dream to be able to play with them but I'm obviously here for a reason. I worked hard at Palace to get the move I wanted and I just want to prove to people that I deserve to be here."