Genesio saw his side beaten 2-1 at FC Nantes on Friday to leave them in third place, but they opened the door to their rivals behind them, who can narrow the points gap uncomfortably close.
Olympique de Marseille moved to within five points with their win against Nîmes Olympique on Saturday while arch-rivals AS Saint-Étienne will be just three points back if they beat Girondins de Bordeaux at home on Sunday.
With Lyon fans unhappy with Genesio's tenure, he has decided to bring to an end his three-and-a-half year stint as first-team boss at OL in the summer, and hopes his decision will allow the team to achieve qualification for next season's UEFA Champions League with the help of their supporters.
'Hope everyone understands the message'
"With six games to go, it's not the moment to leave, but things had to be calmed down and a more favourable environment created. For the interest of the club, because I have always thought the club's interest should be put above those of the individual. I hope everyone is going to understand the mesage and that behaviours will change," Genesio told a press conference on Saturday before sending a plea to the OL fans
"They should come and support their team, because if they want to see Manchester City and Barcelona again next season, it's by finishing second or third. And to do that, we need them."
Since replacing Hubert Fournier in December 2015, Genesio has guided the team to second, fourth and third-placed finishes, while he went undefeated against English Premier League giants City in this season's UEFA Champions League en route to the competition's knockout stages.
But after appearing likely to extend his contract beyond its current expiry in June, fan reaction to back-to-back league defeats to Dijon FCO and Nantes, and a Coupe de France semi-final loss to Stade Rennais FC made Genesio feel the negative vibes generated by questions over his own future had to be brought to an end.
'We need support'
"There was a context around myself that influenced results. It's not the only reason of course, but it was necessary to clarify things," he said. "I think there comes a time when the players and the team need a climate other than the one we had against Rennes or Dijon. To achieve a performance, we need support and a positive environment."
The 52-year-old said his squad were emotional when he told them of the decision, but they could still see him at the club next season. Genesio, who has worked at the club he played for since 2005, did not rule out taking on another role at the Groupama Stadium.
"My priority was always to stay here. I haven't changed my mind about that. I think I'm going to take a bit of a rest after this season. I'm not worn out, but that is part of the discussions I'm having with the president about what I can do next season," he said.
"We have spoken about the possibility of staying at the club in a role as yet to be defined. But the most important thing is to finish the season well in a climate that will allow us to achieve our objectives."