Koeman is currently the bookies’ second favourite to land the job at Goodison Park, following Roberto Martinez’s departure.
The Dutchman is apparently on Everton’s wishlist as they look to find a successor for Martinez who can take them forward.
Everton are predicted to have a large deal of money available to them in the summer transfer window, following the investment of Farhad Moshiri.
However, Wilkins still feels that Koeman would be better off staying at Southampton, and waiting for a top four club to come in for his services.
“I think the bigger club for him is Southampton,” Wilkins said. “We think he is looking for a bigger fish. That’s no disrespect to Everton, but he’s done a magnificent job this guy. He brings top quality players in and I think he’s ready for a real top four go.”
Koeman has been with Southampton for two seasons now, and he has done a wonderful job at St. Mary’s.
The 53-year-old has had to deal with losing a number of his key stars at Southampton, but he has ensured that they have finished in the top half of the Premier League in both his seasons in charge.
Despite the links with Koeman, the current favourite to land the Everton job is Frank De Boer.