"Laurent Blanc has contacted the FFF president, Noel Le Graet, to tell him his decision not to ask for another contract," the FFF said in a statement on their website.
France lost 2-0 to holders Spain in the quarter-finals of Euro 2012 last Saturday.
Blanc took over from Raymond Domenech after Les Bleus were eliminated in the first round of the 2010 World Cup, leading them on a 23-match unbeaten run before they lost against Sweden in the group stage of Euro 2012.
Under Blanc, France sometimes played some nice, quick-passing football but once again, two years after the World Cup fiasco, suffered from off-pitch drama.
Deschamps, Blanc's team-mate and captain in the France squad that won the 1998 World Cup and Euro 2000 titles, is the front-runner to take over amid speculation that he is close to leaving Marseille, who he has coached since 2009.
The French federation's executive committee is due to meet on Tuesday when a replacement for Blanc will be discussed.
Blanc has already been linked with the Tottenham Hotspur job after Harry Redknapp was sacked by the north London club after the Premier League season ended.
"If Laurent Blanc leaves it's a shame because he had indicated his wish to continue. But Didier Deschamps is among the very good coaches," former France captain and current UEFA president Michel Platini was quoted as saying in French media on Saturday.
"It is the FFF president who chooses. I do not think we have bad players. Progress has been made, we have potential but it is our (rightful) place (to have been) in the quarter-finals. Anyway, Didier Deschamps does know the team."
France start their 2014 World Cup qualifying campaign against Finland in September.