The Saints boss is set for talks on Thursday, a report claims.
Koeman is set for talks with Everton, a report claims
Koeman has emerged as the front runner to replace Roberto Martinez - ahead of Sevilla coach Unai Emery and compatriot Frank de Boer, who left Ajax last month.
Emery also features highly on the bookies' list of favourites
And Everton are reportedly looking to tempt the 53-year-old north with a contract that would put him amongst the Premier League's highest paid bosses.
Moshiri (L) will reportedly hand Koeman significant funds to spend
Southampton would be reluctant to lose Koeman, however, and are reported to have held talks over a new deal for the Dutchman, who has also been promised funds at St. Mary's.
The report adds that the former Valencia manager has been frustrated by the sales of several key players over the past two summers.
Koeman, who has led the Saints to record sixth and seventh-placed finishes, has only a year to run on his existing contract.
Morgan Schneiderlin (R) is one of several stars Koeman has seen depart St. Mary's