Everton are currently looking for their next manager following the sacking of Roberto Martinez last month.
A number of names has been linked with the role, but it appears that Southampton manager Koeman is at the top of the list.
A move for him accelerated this weekend, with reports suggesting the Blues have offered him a large wage packet and transfer budget to tempt into switching the south coast for Merseyside.
So far the Saints have failed allow the Toffees to discuss terms, but the Express claims they are quietly resigned to losing the Dutchman.
He has now also told the club that he wishes to speak with Everton, which should accelerate his move to Goodison Park.
Indeed, chairman Bill Kenwright is hoping to have a deal concluded by next week, with a compensation fee of £4million being touted for his services.