Fr Stephen Crossan is reported to have sniffed coke through a Â£10 note on a night of drinking in July 2015 in Banbridge, County Down.
He was in a room with Nazi memorabilia and seemed to say "I shouldn't" as he snorted, the Sun on Sunday reported.
The bishop of Dromore said in a statement that he had no knowledge of the incident.
It allegedly occurred in the parochial house last July after a party.
The Sun on Sunday said it happened at what was then Fr Crossan's parish home in the grounds of St Patrick's Church, Banbridge, in July 2015.
Father Crossan told the newspaper that he took the drugs but said: "It was just the one night and that was it."
A source said that a group ended up back with Fr Crossan after a party and found Nazi memorabilia including flags, hats and an eagle with a swastika on a plinth on his mantelpiece.
In his defence, Fr Crossan told the paper that he was no Nazi and that he collected historical items from every country.
He said he had been on sick leave with depression at the time of the video and said he had since left the Church but was being backed by the parish.
In his statement, Bishop of Dromore John McAreavey said that Fr Crossan had asked for and been granted leave from his pastoral duties at Seapatrick parish in May 2015. He said he had been receiving counselling and was considering his future.
At the start of February 2016, the statement said that the priest had asked for an extended leave of absence from the priesthood.
Bishop McAreavey said he was concerned for the priest's health.
Fr Crossan is no longer living at the parochial house.
A Catholic priest caught on video snorting what appeared to be cocaine has taken leave from the priesthood.