The Equality Act, which comes into force in April, is designed to stop discrimination against gay and lesbian couples wishing to adopt a child, but the Church leaders called for an exemption for Catholic adoption agencies on faith grounds.
The emotional battle between Church and state involved even British Prime Minister Tony Blair, who was said to have favored an exemption, risking a revolt by most of his ministers.
But on Thursday Education Minister Alan Johnson, who has responsibility for adoption, said the government, including Blair, saw no case for special treatment.
"I don't see a case for exemption and I don't think the prime minister does," he told BBC radio.
"The case for no exemption has been made very eloquently. The strength of that argument suggests that we cannot introduce legislation to protect gays and lesbians against discrimination and at the same time allow that discrimination to continue."
Blair later told reporters a final decision would be taken next week and that while he favored the right of adoption by gay couples he also wanted to ensure the Catholic agencies felt able to continue their good work.
"I have always personally been in favor of the right of gay couples to adopt. Our priority will always be the welfare of the child," he said. "Both gay couples and the Catholic agencies have high levels of success in adopting hard to place children."
"I am committed to finding a way through this sensitive and difficult issue."
On Wednesday, Archbishop of Canterbury Rowan Williams, the leader of the world's 77 million Anglicans, and Archbishop of York John Sentamu wrote to Blair backing a call by the head of the Catholic Church in England and Wales Cardinal Cormac Murphy-O'Connor for the special exemption.
"The rights of conscience cannot be made subject to legislation, however well-meaning," said the letter.
Murphy-O'Connor's letter to Blair argued that to force Catholic agencies to place children with gay or lesbian couples went against the Church's teachings.
"We believe it would be unreasonable, unnecessary and unjust discrimination against Catholics for the government to insist ... Catholic adoption agencies must act against the teaching of the Church and their own consciences," he wrote.
Murphy-O'Connor said it would be a tragedy if legislation forced the agencies' closure as this could put some 4,000 children awaiting adoption at a disadvantage.
Johnson said the Church leaders' pleas were a minority view and Jewish and Anglican adoption agencies had made no such call.
"I very much hope that the Catholic Church does continue to provide the important service that they do. But if they don't, I think we can ensure that children are not disadvantaged by that," he said.
"There is a minority view from the Catholic Church and we are sensitive to that. We want to try and find a way through," he said, suggesting a transition period before Catholic agencies were forced to comply.
The 12 Catholic adoption agencies in England and Wales handle around one third of all voluntary sector adoptions.
(Additional reporting by Sophie Walker)