In truth, it is hard to see that the Dutchman even has one after he saw his side slump into the relegation places following a desultory second-half collapse against the rampant Gunners.
Everton are mired in their worst start to a league season in 12 years but the current malaise eating away at the club extends far beyond statistics.
Koeman remains relaxed and defiant — without adopting the aggressive, confrontational stance favoured by so many other managers in his position — and his relationship with chairman Bill Kenwright and majority shareholder Farhad Moshiri reportedly remains decent.
But defeat — and another performance like this — in the Carabao Cup visit to Chelsea on Wednesday could well spell the end for Koeman… assuming he makes it to that fixture.
“It’s how you look at the situation,” the Dutchman said. “If you start to think negatively then maybe there is no solution.
“I’m not like that. But okay, once again, there are all the rumours, all the newspapers but that’s normal.
“The team is underperforming, the team is in a difficult situation mentally, with confidence. They are struggling.
“That’s what we need to change and I personally believe that I can change the whole situation but everybody knows how it works in football. The decision is not in my hands.”
How it works — just ask Craig Shakespeare, to name but the latest example — is for zero tolerance from the boardroom, which could well mean Koeman has already exhausted any residual goodwill left over from a strong debut season in charge at Goodison.
“We have played nine games, and played the top five from the Premier League last season,” added the manager. “But this is not a position in the table that’s good for Everton. I understand. I have been working a long time in football, managing a long time. You don’t need to tell me what can happen.”
It had all started so differently for the Toffees, even allowing for an excellent opening from an Arsenal side featuring Mesut Ozil, Alexis Sanchez and Alexandre Lacazette in the same starting line-up for the first time.
On 12 minutes, Idrissa Gueye dispossessed Granit Xhaka in midfield, passed on to Rooney and the former England man planted a superb right-foot shot into the top corner of Petr Cech’s goal — an almost carbon copy of that goal against Arsenal 15 years and three days ago that announced his arrival on the Premier League scene.
That day, his goal was the winner although that looked unlikely to be the case here, even before Nacho Monreal equalised on 39 minutes.
Xhaka’s long-range shot took an unkind deflection off Gylfi Sigurdsson, forcing Jordan Pickford into a block which allowed the defender to convert from the resulting rebound.
Everton actually started the second half more brightly but, perhaps inevitably, were caught on the counter as Sanchez delivered an amazing left-wing cross, Ozil ghosted between Michael Keane and Phil Jagielka and headed in from six yards.
It looked an uphill task for Everton and, when Gueye earned a second yellow card for a foul on Xhaka after 69 minutes, it was an impossible one.
On 74 minutes, Lacazette wrapped it up after more good inter-play between Sanchez and Ozil and a precise cross from the latter, and the only issue that then remained was by how many Arsenal would win. Xhaka had hit the crossbar by the time Aaron Ramsey deservedly scored in the last minute of normal time, following a defence-splitting pass from Jack Wilshere.
And in the five minutes of added time, both teams added to the surreal feeling of the game by adding a goal apiece.
First, Everton substitute Oumar Niasse charged down a poor Cech clearance and saw the ball ricochet into the net before Sanchez was allowed to surge across the face of the Everton area before beating Pickford with an angled shot from the right.
It was no more than the Chilean deserved for an outstanding performance and, if Arsene Wenger could be assured that level of performance from him, Ozil and Lacazette every week, his own situation would be far happier.
“Sanchez was on fire,” said Wenger. “I believe since he came back from injury he has worked very hard.
“Ozil as well. The two of them are very sharp. They were very mobile and showed their individual quality. It is what makes them dangerous against any defender.
“Ozil was superb. Agile, quick, intelligent. He was always at the service of the team with the intelligence of his passing. When he is at that level he is an exceptional football player.”
Pickford 6; Kenny 6, Keane 5, Williams 5 (Davies 45, 6), Jagielka 5, Baines 6; Gueye 6; Rooney 7 (Lookman 74, 6), Vlasic 5, Sigurdsson 5; Calvert-Lewin 6 (Niasse 74, 6).
Subs (not used) Robles, Mirallas, Martina, Klaassen.
Cech 5; Bellerin 7, Koscielny 6, Mertesacker 6, Monreal 7, Kolasinac 7; Ramsey 8, Xhaka 6; Ozil 8 (Coquelin 82, 6), Lacazatte 7 (Wilshere 77, 6), Sanchez 9.
Subs (not used) Macey, Giroud, Walcott, Holding, Iwobi.
C Pawson 5