Wilshere left Arsenal at the end of last season following the expiry of his contract, ending a 17-year stay at the North London side since joining the club's academy at the age of nine.
"It's down to me to prove a point to Arsenal that they were wrong to get rid of me," the 26-year-old Wilshere told West Ham's matchday programme.
Wilshere struggled with injuries and loss of form at Arsenal and spent their 2016-17 campaign on loan at Bournemouth.
He forced his way back into the team last season but was allowed to leave after Unai Emery replaced Arsene Wenger as manager.
"It'll be a strange one for me as I was never able to go there and play when I went out on loan as you can't play against your parent club," Wilshere added.
"But it's one I'm looking forward to... I quite fancy our chances."
West Ham and Arsenal have made tough starts to the new campaign, losing both their matches.
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