The Toffees' 1-0 defeat at home to Burnley on Sunday (1 October) means they enter the international break just two points above the relegation zone. More worryingly for the Dutchman, having picked up four points in the first two games of the campaign, Everton have then only managed to win once in the Premier League and only twice in the last 10 games in all competitions.
Everton have looked extremely ponderous going forward and the former Southampton manager is now the bookies' favourite to be the first top-flight manager to be sacked this season.
However, while recognising he was in a very delicate position, Koeman remained adamant he was the right man to steer Everton out of trouble.
"Of course we have to improve in winning games," he told reporters. "I know that. It [the future] is not in my hands but I try to get the best out of the players. If there is no commitment and no aggression, then that is maybe a reason to think about my future but I thank the players for their commitment against Burnley."
The Merseysiders were booed off after the loss to Burnley - Jeff Hendrick's first half goal gave the Clarets their second away win of the campaign - and according to The Sun the Dutchman has been given just one game to save his job. After the international break they travel to Brighton and Hove Albion before hosting Lyon in the Europa League and then welcoming Arsenal to Goodison Park.
Majority owner Farhad Moshiri has broken his silence over Everton's dismal campaign and given his full back to Koeman, despite reports over his future. Though the former European Cup winner will surely welcome the vote of confidence, such backing in often followed by a swift dismissal upon the next setback.
"These are early days and Koeman has my total support," Moshiri said, according to a statement obtained by Talksport. "We have great fans and they deserve better. We know the honest and objective expectations of our supporters and will not let them down.
On Thursday (28 September), after Everton conceded a late equaliser to draw 2-2 against Apollon Limassol in the Europa League, Koeman admitted his side were "afraid to play football" and acknowledged they would go into the international break on a very downbeat note.
"It is not nice if you have two weeks for an international break and get a defeat, but that is life and you have to accept it," he added. "But don't accept it, and sit back and wait. We will continue and I will try to do the best for my job. All the other questions are not in my hands."
Despite spending over £200m since arriving at Goodison Park, Koeman is reportedly unhappy with Director of Football Steve Walsh and chairman Bill Kenwright for their failure to replace Lukaku in the summer.