Vardy scored 24 goals for Leicester as they shocked the world to win the Premier League ahead of the Gunners last season.
Pires told the Daily Mail: "There is no reason why it couldn't work out for Vardy.
"He scored a lot of goals in the Premier League last season and is now known around the world.
"He has the ability, comes across as very hard-working which Arsenal like, and he's a great finisher.
"There's a lot more pressure with playing for a big club, so it will be a step up."
Pires feels Arsenal have already done smart business in the transfer window in completing a deal for Granit Xhaka.
He added: "Granit Xhaka is a great signing and to be honest, I'm not sure how many more players are needed.
"A lot is said about Arsenal going forward and if you look at goals scored, they were only a few goals off finishing top of that chart, Vardy's addition could help next year."
Pires said: "The league will be more difficult, especially with the arrivals of Pep Guardiola and Antonio Conte, plus you cannot ignore Mourinho's appointment.
"You expect Mourinho to get Manchester United to compete again.
"His rivalry with Wenger was one the most eagerly awaited clashes from the last decade of the Premier League.
"It's a good fight. It may take a while for him to get his side up to scratch so it's a chance for Wenger to fix that head-to-head record."
Arsenal finished second in the Premier League in the campaign just gone, 10 points behind champions Leicester.