Former France captain and coach Michel Platini said the World Cup finals in South Korea and Japan had marked "the end of an era" for the national team.
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I told him about my choice last week and he said it was an excellent idea
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Michel Platini
"Our record over the last four or six years is remarkable. But this World Cup marked the end of an era. We must change," he was quoted as saying in Le Parisien on Tuesday.
According to reports, Platini, former France coach Aime Jacquet and federation president Claude Simonet have already met to decide on a replacement for Lemerre.
Although no announcement is expected until the federation meets to discuss the issue on 5 July, speculation is already growing about who will take over.
"I'm not entitled to make an announcement. President Simonet will announce his decision on 5 July or before if he wants to," Platini, an FFF vice-president, said.
"I told him about my choice last week and he said it was an excellent idea. Whether he will follow my advice I don't know."
French sports daily newspaper l'Equipe said Platini's first choice was Fulham manager Jean Tigana, but Simonet vetoed that appointment.
The former France midfielder has reportedly already declined an offer for contractual reasons.
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I'm certain people will be after my blood if we don't qualify
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Roger Lemerre
Jacques Santini and Alain Giresse, other former team-mates of Platini, are still among the favourites for the job.
Santini resigned as Olympique Lyon coach on 7 May after leading them to their first French title. Giresse is coach of Moroccan club FAR Rabat.
Asked what the France coach's mission would be, Platini told Le Parisien: "To rebuild. New challenges will come quickly with the Euro 2004 and 2006 World Cup qualifiers."
Lemerre, who presided over the worst performance by a defending champion in World Cup history, appears resigned to his fate.
Despite being under contract until 2004, he said during the tournament: "I'm certain people will be after my blood if we don't qualify."
France were eliminated without winning a game or scoring a goal.