The Toffees’ poor start to the season has seen frustrations rise at Goodison Park, and the Star suggests that a defeat for Everton against Arsenal today could spell the end for Koeman.
The Dutchman’s job is hanging by a thread, with calls for him to be sacked growing louder by the game.
And Everton’s board are also said to be considering dismissing head of recruitment Steve Walsh.
Everton invested heavily over the summer transfer window, but they have failed to unearth many stars as yet.
Most of the Toffees’ summer signings have struggled badly since moving to Merseyside, with only the low profile Nikola Vlasic really impressing.
Everton have been in awful form for much of the campaign, with holes seemingly being exposed throughout the squad.
The Toffees are currently in 16th place in the Premier League table, and if Koeman does get sacked, then Walsh reportedly may have to follow the 54-year-old out of the exit door at Goodison Park.