A leading gay rights campaigner is threatening to name the one in ten Church of England bishops described as "secretly gay" by a fellow bishop.

Peter Tatchell warned that "time is running out" for bishops who continue to oppose same-sex marriage. "If bishops don't want to be outed, they'd better make some pro-gay statements soon," he told Christian Today.

His warning came after the Bishop of Buckingham Alan Wilson claimed in a new book that one in ten bishops could be caught in an "episcopal closet" because of a "reactionary clique" of conservative evangelicals with influence in the established Church.

Bishop Alan denounced "duplicity and hypocrisy" in the Church over homosexuality, claiming that around one in 10 bishops could be secretly gay.

He accused the current episcopate of preaching a 1950s "Janet and John" image of human relationships while adopting an "eyes wide shut" approach to homosexuality in its own ranks and the wider church. And, quoted in The Daily Telegraph, he dismissed a recent order banning Anglican clergy from marrying their same-sex partners as unlawful despite its "blustering menacing tone".

The book, More Perfect Union? is the first to state explicitly what has long been rumoured, that several serving bishops are in gay relationships.

Wilson writes: "By 2014 there were said to be a dozen or so gay bishops. By definition, these men are outstanding priests who have managed to navigate the complexities of a structurally homophobic institution well enough to become its iconic representatives. They may well have a bigger investment than others in keeping the closet door tightly shut."

He says the treatment of the Dean of St Albans Dr Jeffrey John, who was forced to withdraw from a nomination as Bishop of Reading, drew the bars "firmly back across the inside of the episcopal closet door."

Christian Today recently disclosed that Dr John is one of four men on the shortlist to be the next Bishop of St Edmundsbury and Ipswich.

Tatchell said: "I commend the Bishop of Buckingham for his candour and for his valued support for the lesbian and gay community. He is a shining light for love and equality - unlike most of the Anglican leadership.

"Given the Church of England's homophobic policies towards gay people and same-sex marriage, there is growing pressure for the outing of bishops who oppose gay equality. Dossiers are being prepared on a number of senior clergy.

"Support for outing clerical homophobes and hypocrites is increasing all the time.... We don't want to out them but if they do not dissent from the Church's discriminatory policies we'll have no option. Time is running out."

The new controversy echoes that of 20 years ago when OutRage!, of which Tatchell was a prominent member, accused ten bishops of condemning homosexuality in public while living secret gay lives. In a placard campaign, the organisation urged the bishops to "tell the truth."

Colin Coward, recently awarded the MBE for his work as director of Changing Attitude, said: "Bishop Alan identifies the difficulty faced by gay bishops in the Church who discuss gay people and tacitly support teachings which maintain prejudice and discriminate against lesbian and gay Christians. Stating that there are 10 or more gay bishops is not to out them. That is up to the bishops themselves and it is something no bishop has ever done voluntarily.

"No bishop should be outed against their will, and Bishop Alan has not outed anyone. And no bishop should be pressured to reveal their sexuality. Being gay in the Church and constructing an identity which makes you acceptable to those who select bishops comes at a cost.

"The charge the bishops face, all bishops, is the charge of hypocrisy and lack of integrity. Bishops ordain and licence LGBTI clergy, knowing that many of them live with their partner. They are coming under increasing pressure from public opinion, inside and outside the Church, to abandon homophobic practice and teaching. Those bishops who are gay have a particular responsibility to argue for change, but are constrained by the need to conform and hide their sexuality to placate minority conservative opinion."

Bishop Wilson, speaking to Christian Today, said the Church had to change otherwise it would become "more and more irrelevant". He added: "We have gone round and round in circles for such a long time. We just carry on having the same Groundhog Day experience. I speak as someone brought up in the best tradition of Janet and John stories in the 1950s. I have come to see gay people are simply people. The whole phenomenon of homosexuality is within nature, not against nature. The Church needs to change."