Cardinal Philippe Barbarin (pictured) has rejected calls to step down and is accused of protecting an alleged paedophile priest during his time in office.
The Archbishop of Lyon has been told to "take responsibility" by Manuel Valls, Prime Minister of France's secular government.
The Prime Minister has joined a campaign demanding that the cardinal resigns.
However, Cardinal Philippe Barbarin said that he had "never, ever, ever covered up the slightest act of paedophilia."
At present, he sits in one of the most prestigious positions in France's Catholic Church, as Primate of the Gauls.
According to The Times, he is being investigated by French police after allegations surfaced that he did not alert police when a priest admitted that he had sexually assaulted young boys.
Fr Bernard Preynat has been under investigation since January on suspicion of assaulting a number of members of his scout group between 1986 and 1991.
It has been reported that the priest admitted guilt and said he told the church in 1990.
He was removed from the scout group, but remained practicing at priest until last year.
One alleged victim, Bertrant Virieux, told The Times: "We were boys ages between 9 and 11, and it always happened in the same spot in the camp.
"Each of us thought we were the only one to be subjected to these acts."
Writing in a letter to Pope Francis, a number of alleged victims, including Mr Viriteux, said that Cardinal Barbarin had "lost all credibility" by not reporting the alleged admission to police.
But speaking publically, Cardinal Barbarin said that the decisions surrounding the priest were not his, but that of his predecessor.
When asked why he allowed Fr Preynat to keep practicing after he was appointed archbishop in 2002, he said the priest had told him he had stopped all abuse.
Cardinal Barbarin is facing another case relating to another priests alleged abuse of a 17-year-old boy in 2009.
The Cardinal denies doing anything wrong in both instances.
