Everton are currently searching for their next manager following the sacking of Roberto Martinez last week.
A number of names have already been linked with the job, with Dutchman Koeman rumoured to be chief among those being looked at.
It seems though, the Blues efforts to tempt him into a move to Goodison Park will be fruitless though as he will reject a move to the club, according to the Mirror.
Instead he will look to stay on at Southampton and see out the remaining year of his current contract with the club.
Beyond that, he will look to make a move to one of the big clubs in Europe to further his career.
Arsenal are chief among his thoughts, with Arsene Wenger only having a year left on his deal and it unknown whether he will remain at the Emirates beyond next summer.
Barcelona could also be an option, with Koeman having been linked to them before now and sure to be an appealing choice for them should Luis Enrique leave at any point in future.