The President of the Episcopal Commission for the Amazon, Cardinal Cludio Hummes, reportedly asked the pontiff to consider the move because there is a shortage of ordained ministers in the Amazon's remote communities.
Catholicism has seen its influence wane in the region as evangelical churches have set up.
The viri probati proposal – which allows men of proven faith to carry out some duties – has been around for decades, but it has drawn fresh attention under Pope Francis, the first ever Latin American pontiff.
He raised the prospect of lifting the celibacy rule in March, when he told German weekly Die Zeit: "We must consider if viri probati is a possibility. Then we must determine what tasks they can perform, for example, in remote communities."
The Pope has also said that the principle of celibacy for priests is a discipline, rather than a dogma.
The church already allows some exceptions to priestly celibacy.
Those in the eastern rite Catholic Church are allowed to be married, as are married Anglican priests who convert to Catholicism.
It is thought that if successful the pilot project could be extended to other parts of the world, including Africa.
However, the move is likely to anger conservatives in the church, who are already furious about the pontiff's decision to allow divorced people who re-marry to receive communion if their priests or local bishop approve.
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