Wilshere, who came through the Arsenal academy, is on loan at Bournemouth, the Gunners’ opponents Sunday.
Due to the terms of his loan agreement, the 24-year-old cannot play against his parent club at the Emirates.
Wilshere will enter the final year of his contract at the end of the season, leaving Arsenal facing the prospect of losing him on a free transfer in 2018, but Wenger is keen for him to sign fresh terms.
‘He can show his loyalty, like Steven Gerrard did,’ said the Frenchman.
‘We want to give Jack a new contract because we believe he will play for Arsenal again, not to protect our value.
‘I let him go this season because we bought Granit Xhaka, we have Mohamed Elneny, Aaron Ramsey. He wasn’t fit enough to be in the position to start games, so I let him go.
‘If Jack comes back to his right level he will play for us.’
Wenger, though, acknowledges he’s taken a risk by allowing Wilshere to leave on loan without signing a new contract.
Wilshere could potentially sign a deal with an overseas club just over 12 months from now, but Wenger says he ignored his better financial judgment for the player’s own benefit.
‘It was a human case more than a calculated financial case with Jack,’ said Wenger.
‘He’s been educated here since he was eight or nine years old, he has gone for years through difficult periods where he didn’t play (because of injury).
‘I felt honestly that it would be unfair for a guy who has been educated here to have another season where he doesn’t play because I have players who are fitter than him. It was not a mathematical decision.’
Original source: Wilshere can be Arsenal’s Gerrard - Wenger