Dr Williams will retire at the end of this year and his successor has yet to be chosen.
Massey University church historian Peter Lineham said New Zealand bishops were holding back on accepting gay priests for the sake of the unity of the wider church.
The issue has been passed to a Commission on Same-Gender Relationships led by former Governor-General Sir Anand Satyanand, which is due to report by 2014.
He said the Auckland council meeting would have to recognise that the covenant had "failed" because it had not received enough support to make it workable.
"I think they are now at the point where they have to come up with a new idea," he said.
"What I think is going to happen is that the links between the different national churches are going to become more informal."
However, the London-based Secretary-General of the Anglican Communion, Canon Kenneth Kearon, said the issue was unlikely to be resolved until the next council meeting in 2015.
"Of the 38 provinces, three have said no so far, about 12 to 15 have adopted the covenant, and the vast majority are still talking about it," he said.
"I don't think we will begin to come to a conclusion for another three years."
Instead, he said, the major issue on the agenda in Auckland would be gender-based violence, an issue the council has picked up from its women's and family networks.
The Rev Charles Waldegrave of Lower Hutt's Family Centre, who is co-ordinating a public forum on the issue to be chaired by Dr Williams at Auckland's Holy Trinity Cathedral at 7.30pm on Tuesday, said it was a "breakthrough" to get family violence on the agenda.