Everton have remained silent on their search for a new boss, but it is believed Koeman has always been the first choice of new major shareholder Farhad Moshiri.
The list of candidates also included former Manchester City manager Manuel Pellegrini, Sevilla’s Unai Emery and former Everton manager, David Moyes.
Koeman, 53, was appointed manager of Southampton in 2014 and led the club to seventh in the Premier League in his first season in charge.
This year, they finished in sixth place, again qualifying for the Europa League.
On the final day of the season Koeman indicated he would require assurance that Southampton was showing ambition before committing himself to extending his deal.
Koeman: “It is important to know what we can do to be stronger because in my opinion that’s ambition. If you don’t have that ambition, you start to be sloppy and that is not what I like to be. It is possible [to sign a contract next week] but that is always depending on the conversation.”
Southampton chairman, Ralph Krueger, had previously indicated that he would not stand in Koeman’s way if he wanted to leave.
Everton will look to appoint a director of football to work alongside Koeman – likely to be Sevilla’s Monchi, who confirmed on Friday that he intends to leave the Europa League winners this summer.
Monchi: “For about a month I noticed that something wasn’t right in my head. I figured it was time to stop and make a change. My doubts then returned but late on Sunday I made the decision [to leave]. Still, what I can say is that I won’t pay my release clause. I won’t leave Sevilla through the backdoor.”