The Arsenal boss admitted that he is a fan of the Kenyan, but insisted that he is not planning on trying to lure him away from St Mary's next month.
"I rate the player but we are not on the case," he said.
Despite his insistence that the former Celtic man will not be on his way to north London this winter, he did hint to Sky Sports that he is considering doing some business during the window.
"We will see what is there. We have a whole December to go first," he added.
With Francis Coquelin out until at least the new year with injury (as reported by the Telegraph), the Gunners are temporarily left with just one holding midfielder that is happy to sit just in front of the back four in Mathieu Flamini.
And while the Frenchman has been in good form for much of the campaign, Wenger will know that another injury could leave the defence exposed.
Wanyama has performed the same role as Coquelin for Ronald Koeman's side. And while the Saints have failed to hit the heights of last season during this current campaign, it has not been due to a lack of performances from the 24-year-old.
Arsenal fans will be hoping that given the influence of Coquelin since his emergence last year, that, even if it is not Wanyama who comes in, Wenger uses the January window to strengthen in that area of the pitch as it may prove to be the difference between whether they win the title, or come up agonisingly short.