The Toffees have now won just twice in their past 12 games and are rooted to the bottom of Europa League Group E after picking up just one point from their opening three matches in the competition.
They appeared to be heading for a point after Ashley Williams had cancelled out Nabil Fekir's early opener, but Bertrand Traore flicked home with 15 minutes remaining to condemn Koeman's side to yet another defeat.
The beleaguered Blues welcome Arsenal to Goodison Park on Sunday in a must-win game, but Koeman has cast doubt on his future by admitting that the board might well decide to get rid of him.
"Of course, if the team is not winning and the team is not performing but maybe they can, then it's always the manager," he told BT Sport .
"The final decision is by the board, it is not by myself. I'm strong, I'm trying to do everything and if the board thinks I'm not that right man on this position then they will tell me."
The growing frustration on Merseyside was summed up by Williams sparking a second-half brawl between both sets of players after a reckless shove on Lyon goalkeeper Anthony Lopes, with fans unnecessarily lashing out at visiting players in the ensuing melee.
Koeman attributed Williams' involvement to frustration, but was quick to deflect attention towards his dissatisfaction with the referee's performance.
"That was a certain irritation in that moment by Ashley because he's that player who likes to win, who cannot deal with a lot of disappointments," he added.
"But I think the mistake was also by the referee. They made a lot of theatre and he did nothing. Fekir was I think 10 times on the floor diving, not even a yellow card and that has grown our irritation.
"Even extra time - it's five minutes. We had six substitutions, that's three minutes. We had the fight, maybe two minutes. And they had I think eight times injuries in the game and it's five minutes? Come on please."