However, his suggestion goes directly against the rules of the Church of England and the Roman Catholic Church, which state that only the union of a man and a woman can be celebrated by a priest in church.
It comes as the Government is pushing through an Equality Bill that religious groups fear will force them to give jobs to homosexual youth workers or secretaries, even if their faith maintains that same-sex relationships are sinful.
Mr Bryant, a former deputy leader of the Commons who is now a junior Foreign Office minister, trained as an Anglican priest and served as a curate before becoming the MP for the Rhondda.
He was questioned by the magazine Time Out this week on whether he believed civil partnerships, introduced to England and Wales in 2005, should be scrapped in favour of same-sex marriage.
The minister replied: "All my friends who have entered into a civil partnership refer to it as their 'marriage' or their 'wedding' so the most important issue is that nobody should be discriminated against because of their sexuality.
"I would like to see churches be much more open to the idea of gay relationships or partnerships being celebrated in church."
Nick Herbert, a Conservative MP who is in a civil partnership, and Stephen Williams, who was the first openly homosexual Liberal Democrat MP, both told the magazine that they would support the legal introduction of same-sex marriages.
Church of England rules state that civil partnerships are not equivalent to marriage and cannot be celebrated in church, and the battle between conservatives and liberals over homosexuality has driven the worldwide Anglican Communion to the brink of schism.
Last summer a rector was strongly criticised by the Church of England's two archbishops after he conducted a "wedding" service, including traditional liturgy and the exchange of rings for two homosexual priests, at his 12th century church in the City of London.
The Rt Rev Michael Nazir-Ali, the Bishop of Rochester, said: "Of course all citizens must have equal rights without discrimination. But marriage is the basis of the family, and the stability of the family is grounded in the sameness in difference between men and women.
"Those who make public law have to realise that people of faith have consciences that need to be respected."
A spokesman for the Church of England said: "The Church of England's approach has always been clear: marriage is the lifelong union between a man and a woman, and that is what the liturgy of the C of E Marriage Service is exclusively intended for.
"On civil partnerships the Church continues to uphold that standard, to affirm the value of committed, sexually abstinent friendships between people of the same sex and, at the same time, to minister sensitively and pastorally to those Christians who conscientiously decide to order their lives differently.
"Some who register civil partnerships seek recognition of their new situation and pastoral support by asking members of the clergy to provide a blessing for them in the context of an act of worship. The Church expresses what it believes through the liturgy of its worship. As there is no theological consensus about same sex unions, no such liturgy is authorised."
Mike Judge, a spokesman for The Christian Institute, a campaign group, said: "Churches are open and welcoming to all people, but that is not the same thing as forcing churches to celebrate behaviour which conflicts with their religious ethos. It would be like forcing the Labour Party to celebrate a Conservative election victory. Surely the world is big enough to allow people to be free to disagree."
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