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 shortly after having initially accepted it, following protests from traditionalists.
He was also a candidate for Bishop of Southwark in 2010 but was rejected. Evidence emerged that this was because of his sexual orientation.
The Church of England has already agreed to allow people in civil partnerships to become clergy, provided they promised they would remain celibate.
BBC religious affairs correspondent Robert Pigott said that, given the tension surrounding the issue of sexuality, the Church’s decision to allow men in civil partnerships to become bishops represented a major concession and one with considerable symbolic significance.
Conservative evangelicals have warned they would be willing to bring in bishops from overseas to avoid serving under a gay bishop.
The Reverend Rod Thomas, chairman of the evangelical group Reform, said the idea of appointing people in civil partnerships as bishops had not been agreed or debated by the wider Church.
“That would be a major change in church doctrine and therefore not something that can be slipped out in the news, it is something that has got to be considered by the general synod.”
He said there would be great divisions in the Church if clergy in a civil partnership were appointed as a bishop.
Worldwide, anger over the appointment of actively gas men and women as bishops, especially in the US, has stimulated the Gafcon movement, through which conservative Anglican provinces in Africa and elsewhere have begun to function independently of the official Anglican Communion.
-www.bbc.co.uk, 4 January 2013