In May, he was given a one-year sentence for failing to report allegations of abuse by a priest in the 1970s. Wilson is the world's most senior Roman Catholic cleric to be convicted of the offence.
Earlier this month, Australian Prime Minister Malcolm Turnbull called on the Pope to sack him, BBC reported on Monday.
During his trial, Wilson denied any knowledge of abuse committed by priest James Fletcher during his time as an assistant priest in Maitland, 130 km from Sydney.
Fletcher was convicted of nine child sexual abuse charges in 2004, and died in jail in 2006.
However, the magistrate found that Wilson had been told about the abuse by the young victims and that he had failed to report the crimes because he wanted to protect the Church's reputation.
Wilson has said he will appeal against his conviction. He could avoid jail after the court ordered he be assessed for "home detention".
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