Pope Francis did say the Ordinariate for ex-Anglicans was unnecessary - but that does not mean it will be Papal policy, an Anglican bishop has said.
Bishop Gregory Venables told the BBC the published account of Cardinal Bergoglio's remarks to him was correct.
A spokesman for the UK Ordinariate commented that it was set up with the highest Catholic legal authority.
A Vatican spokesman said last month the Ordinariates were "a permanent feature" of the Catholic Church.
The Ordinariate of Our Lady of Walsingham was set up in the UK in 2011 in a move strongly supported by the last Pope, Benedict XVI, to allow discontented former Anglicans a place within the Catholic Church where they could preserve elements of their "Anglican patrimony".
It was followed by two other Ordinariates, covering North America and Australasia.
Bishop Venables, who is the Anglican Bishop of Argentina, told the BBC News website that Cardinal Bergoglio, then Archbishop of Buenos Aires, had invited him to breakfast in 2009 when the Ordinariate was first suggested.
Remarks to friends
In Bishop Venables' words as published by the Anglican Communion News Service, "he called me to have breakfast with him one morning and told me very clearly that the Ordinariate was quite unnecessary and that the Church needs us as Anglicans."
Bishop Venables told the BBC News website that the quotation of him was accurate, but had not been meant for publication and had appeared on the Anglican Communion website without his consent.
He said he had merely made some remarks to some friends which he had not circulated widely, and added that he did not believe the remarks would reflect the future position of the Catholic Church.
Pope Francis, "as Pope, will assume a position which is balanced, and which honours what others have said," Bishop Venables stressed.
Asked about the remarks, Vatican spokesman Father Federico Lombardi said that ties between the two churches had been strengthened in recent years, especially during Pope Benedict's visit to Britain, and that they looked likely only to get stronger still.
Last month Fr Lombardi told the Catholic Herald that the establishment of the ordinariate was a project, "particularly close to the heart of Pope Benedict XVI."
He said: "The ongoing development of these structures in the future will be a lasting legacy of his pontificate, but also a continuing contribution to the work of Christian unity and ecumenism.
He also said "The Personal Ordinariates are a permanent feature in the life of the Church and a sign of our lasting and unswerving commitment to that ultimate goal".
A spokesman for the Ordinariate said the words that had appeared were those of Bishop Venables, not the Pope.
"The Personal Ordinariates were established by an Apostolic Constitution, the highest form of legal document in the Catholic Church," he said.
"We welcome the election of our new Holy Father and rejoice to be in the full communion of the Church, whilst also bringing the great gifts of our Anglican heritage as a treasure to be shared with all."
In his published remarks, Bishop Venables also said of the new Pope: "He is much more of a Christian, Christ-centred and Spirit-filled, than a mere churchman. He believes the Bible as it is written.
"I have been with him on many occasions and he always makes me sit next to him and invariably makes me take part and often do what he as cardinal should have done.
"He is consistently humble and wise, outstandingly gifted yet a common man. He is no fool and speaks out very quietly yet clearly when necessary."