The husband of a woman thought to have had an affair with a Catholic bishop is threatening to sue the Church.

The husband, who remains nameless, has branded Bishop Kieran Conry of Arundel and Brighton a "hypocrite", telling the Mail on Sunday that the clergyman's behaviour has been "appalling".

"He is someone capable of creating that emotion and distress without taking any interest in the effect on those going through this," the man said.

"It smacks of arrogance...The bishop is supposed to set the best example for a lot of people. To think that this is a person who people turn to for marriage advice is unbelievable."

The man's lawyer said he is considering legal action against the Church amid allegations that "they've known for years that the bishop has been having affairs".

"If they'd taken action he [the husband] almost certainly would not have lost his marriage and his children would not be having to be brought up in a broken family," Claire Kirby said in a statement.

Bishop Conry, 63, first assumed his position in 2001, but resigned this weekend after confessing to having had relationships with parishioners.

"I am so sorry to confess that, going back some years, I have been unfaithful to my promises as a Catholic priest. I would like to reassure you that my actions were not illegal and did not involve minors," a statement read out in all churches in his diocese on Sunday said.

"I want to apologise first of all to the individuals hurt by my actions and then to all of those inside and outside the diocese who will be shocked, hurt and saddened to hear this.

"I am sorry for the shame that I have brought on the diocese and the Church and I ask for your prayers and forgiveness."

The bishop has denied that the relationship with the woman in question was sexual, and says his resignation comes instead in response to a separate relationship from six years ago.

However, the Mail on Sunday claims to have seen handwritten love letters between Bishop Conry and his 43-year-old parishioner.

The newspaper says the affair, though possibly not sexual, lasted over a year, and involved the woman spending at least three nights a week at the bishop's house in West Sussex.

The former clergyman told the paper's reporters that it is a "relief" to no longer have to keep his transgressions a secret.

"In some respects I feel very calm. It is liberating. It is a relief," he said.

"I don't think it got in the way of my job, I don't think people would say I have been a bad bishop. But I can't defend myself. I did wrong. Full stop."