The archbishops of Canterbury and York are planning a bid to stop a Church of England split over women bishops.
Draft legislation on women bishops will be debated by the Church's governing body, the general synod, next month.
Dr Rowan Williams and Dr John Sentamu will propose a new concession to opponents of women bishops.
It involves "co-ordinate jurisdiction" between a woman bishop and another bishop caring for traditionalist parishes in her diocese.
The proposal by the two most senior figures in the Church of England is aimed at catering for objectors in dioceses where a woman is consecrated a bishop.
The question of the powers of a bishop appointed to look after parishes which do not accept women bishops have been the clearest obstacle to a compromise on the issue.
Opponents do not want bishops appointed to look after them to obtain their powers by delegation from a woman diocesan bishop.
On the other hand, supporters of women bishops say that if the bishop caring for traditionalist parishes holds his powers independently, it will impair the woman bishop's authority in her own diocese and make women "second-class" bishops.
Precise text awaited
The latest move comes two years after the general synod rejected ideas put forward by opponents of women bishops, such as new dioceses or a special class of bishop.
Instead, the general synod said women bishops should have the power to make local arrangements for objectors if needed.
Traditionalist grouping Forward in Faith backed the archbishops' move.
It said: "Forward in Faith warmly welcomes today's statement from the archbishops of Canterbury and York and now looks forward with great interest to seeing the precise texts of the amendments to the draft measure which they will propose to the general synod next month."
However, Hilary Cotton, vice-chairwoman of Watch (Women and the Church), said major concessions had already been made by those in favour of women bishops.
She said: "We cannot give an immediate response to whether we can support this amendment. But I would want to say that supporting the legislation as it is drafted is a significant compromise from us."
Objectors to the appointment of women bishops fall broadly into two groups.
Some Anglo-Catholics, who deeply revere Church ceremonies and are close to Roman Catholic practices, believe women are precluded by the Church's teaching from being priests or bishops and that appointing them will impair relations with Rome.
Many of the Evangelicals within the Church, who place great stress on following the teachings of the Bible, believe that the Scriptures require "male headship" in families and in the Church.