The French manager signed Walcott when he was just 16 years old and is now looking for the winger to show some loyalty to the club amid speculation that he may decide to leave on a free at the end of the season.
Walcott has been in negotiations with the north London club for some time as they attempt to persuade him to sign a new contract and commit his future beyond the end of the current campaign.
It has been suggested that Walcott is looking to secure a deal worth around £100,000 a week which would make him one of the top earners at the club.
Arsenal have been unwilling to match his demands but Wenger wants the England international to repay the faith shown in him when he was signed from Southampton and immediately given a chance in the first team.
“What is important to keep the young players together is that at some stage I have stood up for them; I played you at 17, 18, 19 years of age and I know I took a gamble,” said Wenger, as quoted on Sky Sports.