The announcement would allow gay clergy to become bishops if they promise to be celibate.
Groups representing gay Anglicans have welcomed the move but conservative evangelicals have called it “divisive”.
Some critics say it is undermining church teaching about homosexuality in the hope of winning public approval.
The issue has split the church since 2003 amid a row over gay cleric Jeffrey John becoming Bishop of Reading.
Mr John, now Dean of St Albans, was forced to withdraw from the role of Bishop of Reading shortly after having initially accepted it, following protests from traditionalists.
The Church of England has already agreed to allow people in civil partnerships to become clergy, provided they promised they would remain celibate.
The BBC’s religious affairs correspondent Robert Pigott says the latest decision to make the same provision for bishops has reignited Anglicans’ most deep-seated and destructive dispute.
The Lesbian and Gay Christian Movement said although it was a step forward, gay bishops should not have to be celibate.
Ruth Hunt, director of public affairs for gay rights campaigners Stonewall, said: “I’m sure celibate gay men will be thrilled by this exciting new job opportunity, if perhaps somewhat perplexed as to how it will be policed by the Church.”
The Rev Colin Coward, director of the Changing Attitude group, which campaigns for the full inclusion of gay, lesbian, bisexual and transgender people in the Church, said the statement “will be laughed at by the majority in this country,” and added that insisting on celibacy was wrong.
Christina Rees, a member of the Synod and the Archbishops’ Council, said it was “good news” for gay male clergy, but highlighted the continuing lack of female bishops.
Conservative evangelicals denounced the concession outright and insisted that few people believed clergy in civil partnerships were genuinely celibate.
In a statement, Michael Lawson, chairman of the Evangelical Council of the Church of England, said: “At the very least [it] will spread confusion and at worst will be taken as an effort to conform to the spirit of the age.”
And the reverend Rod Thomas, a spokesman for Reform, an evangelical network in the Church of England, said: “It’s a very worrying development.
“If someone were to be appointed who was in a civil partnership, that would be a very divisive step, both within England and across the Anglican Communion.
“Although the Church says they would be required to declare that they are celibate as part of their appointment, the fact is that this is unenforceable.”
Meanwhile, Canon Chris Sugden from Anglican Mainstream said: “Since a decision to move from the current position would be a grave departure from the Church’s doctrine and discipline; it should be made by Bishops in Synod not by Bishops alone.”