Paul Merson believes Roberto Martinez is in his final weeks as Everton boss, and has named the manager he expects the club will target to replace him with.
He said: "I don't think it will be long [until Martinez goes]. But he's a good manager, he will do well abroad. I think they may go for someone like Koeman."
Koeman has been at Southampton for two seasons, and has guided them to successive top 10 finishes.
Everton have just received a huge cash investment from Iranian billionaire Farhad Moshiri, and this could tempt Koeman.
The Mirror reported this week that Koeman has just one year left on his Southampton contract and has not shown an inclination yet to renew the deal, and named Everton as one of two clubs, along with Arsenal, who are monitoring the situation.
With Arsenal showing no signs of sacking Arsene Wenger despite supporter dissatisfaction, Everton could be in pole position to lure the Dutchman away if they vacate their managerial position by dismissing the under performing Roberto Martinez.