But the Vatican issued a statement saying some parts of a long article in the left-leaning La Repubblica were not accurate, including one that quoted the pope as saying that there were cardinals among the abusers.
The article was a reconstruction of an hour-long conversation between the pope and the newspaper's founder, Eugenio Scalfari, an atheist who has written about several past encounters with the pope.
"Many of my collaborators who fight with me (against paedophilia) reassure me with reliable statistics that say that the level of paedophilia in the Church is at about two percent," Francis was quoted as saying.
"This data should hearten me but I have to tell you that it does not hearten me at all. In fact, I think that it is very grave," he was quoted as saying.
The pope was quoted as saying that, while most paedophilia took place in family situations, "even we have this leprosy in our house".
According to Church statistics for 2012, the latest available, there are about 414,000 Roman Catholic priests in the world.
The Vatican issued a statement noting Scalfari's tradition of having long conversations with public figures without taking notes or taping them, and then reconstructing them from memory. Scalfari, 90, is one of Italy's best known journalists.
While acknowledging that the conversation had taken place, Vatican spokesman Father Federico Lombardi issued a statement saying that not all the phrases could be attributed "with certainty" to the pope.
Lombardi said that, in particular, a quote attributed to the pope saying cardinals were among the sex abusers was not accurate and accused the paper of trying to "manipulate naive readers".
Last week, the Argentine pope held his first meeting with victims of sexual abuse by priests.
He told them the Church should "weep and make reparation" for crimes that he said had taken on the dimensions of a sacrilegious cult. He vowed zero tolerance for abusers and said bishops would be held accountable if they covered up crimes by priests in their diocese.