The England international moved to Bournemouth on loan, on transfer deadline day after struggling with injuries over the past 18 months.
The 24-year-old is in search of regular football to regain his fitness and competitive edge.
He played only three games for the Gunners due to injuries last season, with Wenger insisting his youth academy product is “world class”.
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“He is a world-class player, he doesn’t have to show me his quality,” said Wenger.
“I hope personally that he stays his whole career here.”
The Arsenal boss further revealed he sanctioned Wilshere’s loan move after the midfielder expressed concern to feature on a regular basis for the club.
“We had a chat together,” added the Frenchman. “He felt ready to play.
“I couldn’t guarantee that and it looked logical to give him time – the challenge is to play the games.”
Bournemouth are seeking their first Premier League win of the season as they host West Bromwich Albion on Saturday, with Wilshere expected to feature.
Wilshere is expected to feature for Bournemouth on Saturday as they seek their first win of the season when they host West Brom.