The Church of England Synod is discussing a compromise plan for the creation of female bishops put forward by its two most senior figures, in a last-minute attempt to avoid a damaging split.
The proposal, from the Archbishops of Canterbury and York, would exempt traditionalist parishes from serving under a woman bishop.
But liberals in the synod have said they will not accept any measure that dilutes women bishops' authority, the BBC reports.
Some of the traditionalists have threatened to convert to Roman Catholicism over the issue of female bishops.
At the centre of the row are plans for "extra" or "nominated" bishops appointed to look after parishes that do not agree with the appointment of women.