Everton boss Ronald Koeman said the 2-1 Europa League defeat at home to Lyon on Thursday puts his job under greater scrutiny.
The Toffees remain bottom of Group E, without a victory in the group stage, and without a win in four games in all competitions, the BBC reports.
Ashley Williams looked to have earned Everton a draw moments after sparking a scuffle, but Bertrand Traore flicked in a clever winner for the French side.
"The final decision is by the board not myself," Koeman said of his position.
"If the board thinks I am not the right man they will tell me."
Nabil Fekir put the visitors ahead at Goodison Park with an early penalty, but there were signs of improvement from an underperforming Everton in a heated clash on Merseyside, with substitute Gylfi Sigurdsson striking the post with a free-kick at 1-1.
Everton have only one point after three of their six group games. They are six points behind leaders Atalanta and four behind second-placed Lyon, with a crucial trip to France to come in November.
Asked if his position defeat increased the scrutiny on his position, Koeman said: 'Of course. If the team is not winning and if the team is not performing when maybe they can, then it's always the manager."