The general synod - the Church's parliament - will be urged to adopt a similar policy to other bodies which forbid BNP membership, like the police.
The move comes after the leaked publication of the names of 12,000 BNP members in November.
The list contained five "Reverends" but the Church said none was a licensed or serving clergy member.
The Association of Chief Police Officers policy states that no member of the police service may be a member of an organisation whose constitution, aims or objectives contradict the general duty to promote equality.
It specifically mentions the BNP as one such organisation.
At the meeting of the synod next month one of its members, Vasantha Gnanadoss - who works for the Metropolitan Police - will submit a private members motion calling for a similar policy to apply to all clergy, candidates for ordination and lay persons speaking on behalf of the Church.
She said the policy would make it more difficult for organisations like the BNP to exploit the claim that there are members of the Anglican clergy that support them.
"Of specific relevance to this motion are some of the tactics adopted by the BNP, which in recent years has sought to identify itself as Christian and sometimes specifically with the Church of England, in order to further its agenda," she said.
William Fittall, secretary general of the general synod, said it was already Church of England policy that people should not enter ordained ministry if they held racist views.
He added, however, that it would be harder for the Church to enact a formal policy aimed at the BNP.
"Not long ago the synod passed the Clergy Discipline Measure, which specifically said you could not discipline a member of the clergy for political views or membership of a political party," he said.
A BNP spokesman said the party was aware of the efforts of Ms Gnanadoss and denied it was racist.
"There are members of the general synod who are sympathetic towards us," he said.
"This is a disgraceful way to politicise the Church. The Church has got far more important things we feel to worry about... rather than a vindictive campaign against a perfectly legitimate political party".