The motion points to the policy used by the Association of Chief Police Officers, which prevents police officers from becoming members of the BNP or other organisations whose aims contradict the duty to promote race equality.
The BNP campaigns for the voluntary repatriation of immigrants and for the interests of indigenous people of Britain. The party has been charged with being extreme and racist, although its leader Nick Griffin denies this charge.
The motion was put forward by Vasantha Gnanadoss, who works for the Metropolitan police. She claims the motion is needed to prevent parties like the BNP associating themselves with the Church. A ban on membership would also send a clear message against racial prejudice, she claims.
Until now there have been no rules banning membership of any political party for clergy, although in the past Church of England leaders have encouraged voters to stay away from the BNP during elections.
The Church of England has also previously passed a resolution against the “sin of racial prejudice”.
Last year, a list of 12,000 BNP members was leaked to the public, which included five members of the Church of England clergy. The Church noted at the time that none of them were serving Anglican priests.
Speaking to the BBC, Canon Ivor Smith Cameron, former chaplain to the Queen, said that "very, very few" members of the clergy had joined the BNP.
He said that he supported the current proposal for a ban on membership, saying, "One of the reasons for my existence as a clergyman is to preserve racial and community harmony and the work of the BNP clearly does not do that.
"Furthermore, in its websites, its blogs ... it is proclaiming itself to be a Christian organisation and I resent that deeply."
Alison Ruoff, a lay member of the Synod, said such a ban was wrong in a “free country”.
"I think it's totally over the top, totally unnecessary. I want to trust clergy. The more you say, 'Thou shalt not,' the more people want to," she said.
"And of course, even if you don't join it, anybody, but anybody, including clergy, can vote BNP without anybody knowing."
The Church of England General Synod got underway on Monday at Church House, Westminster, with an address from the head of the Roman Catholic Church in England and Wales, Cardinal Cormac Murphy-O'Connor, on the prospects for communion between the two Churches.
In his first and almost certainly last address to Synod before his retirement, the Cardinal said the Anglican and Catholic Churches needed to continue working towards the goal of full communion "even if it still seems so distant".
He went on to express his regret over challenges to unity among Anglicans.
"Divisions within any church or eccclesial community impoverish the communion of the whole Church. We Roman Catholics cannot be indifferent to what is happening to our friends in the Anglican Communion and, in particular, in the Church of England,” he said.
In the coming days, Synod will debate the uniqueness of Christ, a draft Covenant to mend rifts in the Anglican Communion, human trafficking, the impact of the financial crisis, and the damaging effect of rising water bills on many parishes.
