Priests in the eastern rite Catholic Church are allowed to be married, as are married Anglican priests who convert to Catholicism.
He said the Church should consider allowing viri probati - married men tested to be of exemplary religious faith and virtue - to be ordained as priests. "But then we must also consider what tasks they could perform, for example in isolated communities".
Despite that aforementioned "enormous" priest shortage, Pope Francis did not mention opening the priesthood up to women. Now the pope is questioning the church's longstanding rule that priests can not be married.
While saying "optional celibacy" is not the solution, Pope Francis expressed an openness to studying whether so-called "viri probati" could be ordained.
In 2014, Bishop Erwin Krautler, bishop of Xingu in the Brazilian rainforest, told the Austrian newspaper Salzburger Nachrichten that he spoke to the pope about the desperate shortage of priests in his region.
Roman Catholic clergy have been expected to remain celibate since the 12 century when the Second Lateran Council in 1139, definitively passed a rule forbidding priests to marry. But CNN points out this isn't the Pope opening up the priesthood to marriage.
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Kaczynski has long said member states should be given more power within the bloc at the expense of European Union institutions. A Downing Street source said: "We've always been clear he was doing a good job, and that we supported him".
It is important to note that this option would apparently only be open to men who are already married, and that single men who are already priests would still not be allowed to marry.
Another cause of concern are priests leaving their duties to get married.
The decision is said to have been prompted by a shortage of priests in some remote areas.
Despite his remarks on the limited possibility of allowing married men to enter the Church, the Pope ruled out dropping celibacy as a requirement for the priesthood. In the United States, the celibacy requirement is partly what sets priests apart from clergy in Protestant denominations.
There are some exceptions already.