The Saints are said to be preparing a January bid for the England international, who is understood to be willing to take a pay cut in order to gain playing time elsewhere ahead of this summer's World Cup.
Despite using Walcott for just a combined 50 minutes in the Premier League this season, though, with all five of his appearances coming from the bench, Wenger does not want to lose the versatile attacker.
"I want him to stay," he told reporters. "I heard about [the Southampton rumours], but personally I want him to stay."
Walcott has also been used eight times in the Europa League and EFL Cup, scoring four goals and setting up two more for his teammates.