While he has previously spoken of the need for better screening of candidates for the religious life, his comments, in a new book, suggesting priests who cannot keep their vows of celibacy should leave are some of his clearest to date.
Pope Francis made the comments in a book-length interview with Spanish priest Fernando Prado, called The Strength of Vocation, in which he discusses the challenges of being a priest or nun today.
Pope Francis said in the book that homosexuality in the Church "is something that worries me".
It is due to be published this week in several languages.
"The question of homosexuality is a very serious one," he said, adding that those entrusted with training men to be priests must be certain candidates are "humanly and emotionally mature" before they can be ordained.
This also applied to women who wanted to enter female religious communities to become nuns.
In the Catholic Church, priests, nuns and monks all take vows of celibacy.