Lyon, French champions from 2002-08, spent about 30 million euros to recruit France playmaker Yoann Gourcuff, striker Jimmy Briand and defender Pape Diakhate and yet they lie second from bottom on five points.
"I am the right man for the situation," coach Claude Puel told reporters. "I am not going to resign."
Saturday's home defeat to St Etienne, their first in 16 years in French football's fiercest derby, was one too many for the fans, who called for Puel to resign.
Almost two hours after the game, some 2,500 of them were still protesting in the north kop as the club slipped into the relegation zone for the first time since 1995 despite having the league's biggest budget.
President Jean-Michel Aulas went to them and said: "I told you last week that we would assess the situation by the end of October. If you give me one month to deal with it I can promise you that we will be soon back on top of the bill.
"Give the club a chance. I'm not saying a chance to Claude Puel. I'm not saying he's the man for the situation but right now, that's all we have."
On Saturday, Lyon dominated throughout, hitting the woodwork three times, only for the league's top striker, Dimitri Payet, to find the back of the net with a fine 75th-minute free kick.
Payet also cleared off his own line twice in a tense encounter that Lyon would probably have deserved to win.
"Firing Claude Puel after such a great game would be unfair," Aulas told reporters.
"If we continue to play like this, we are going to extricate ourselves from this situation."
Next for Lyon is a trip to Hapoel Tel Aviv in the Champions League on Wednesday.
