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 billionaire who acquired a 49.9% share in the club in February.
The list of candidates also includes former Manchester City manager Manuel Pellegrini and Sevilla’s Unai Emery.
David Moyes, who left Everton after an 11-year tenure as manager to succeed Sir Alex Ferguson at Manchester United in 2013, has also been spoken to.
Moshiri also wants to appoint a director of football as he moves towards a more continental management model, but attempting to secure former Netherlands and Barcelona defender Koeman has always been his top priority.
Koeman, 53, was appointed Saints manager 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.
Speaking about his future last month, he said: “I am manager of Southampton. Everyone knows I have one year more on my contract.”