The Mediapartwebsite names all 25 bishops, five of whom were still in office in January, and accuses Lyon's Cardinal Philippe Barbarin of having known about abuse by five priests without notifying police.
Several cases of covering up sexual abuse against Barbarin were dismissed last June.
Among the other bishops named by Mediapart are Besançon Archbishop Jean-Luc-Bouilleret, Bayonne Bishop Marc Aillet, Yves Le Saux of Le Mans and Mgr Bernard Fellay of the Society of Saint Pius X, a traditionalist fraternity that is going through a reconciliation process with the Vatican.
Of the 339 victims, 288 were minors, the site says.
In a joint probe with the Cash Investigation TV programme, Mediapart claims that 90 priests have been moved because of accusations of paedophilia.
Following a major scandal involving Bishop Pierre Pican in 2000, the French Catholic bishops' conference ordered its members to report cases of abuse of minors to the legal authorities.
But half of the cases reported by Mediapart are alleged to have been reported since then.
The church refused to take part in the Cash Investigation programme, which is to be broadcast on Tuesday, complaining that interviews were conducted in a spirit of accusation rather than explanation.