According to the Mail on Sunday the idea was formulated by bishops and will be discussed by the Church of England's General Synod next month.
Since 2005 gay clergy wanting to enter civil partnerships have had to vow to they will remain celibate. This is worded in canon law as being "willing to give assurances to his or her bishop that the relationship is consistent with Church teaching".
Clergy wishing to become bishops must make similar promises to their archbishop.
An un-named source, identified as a 'senior conservative', told the paper: "This is a slippery slope.
"It will mean that gay clergy will have even less incentive to remain celibate than they do now.
"I fear the next step will be gay marriage."
The worldwide Anglican Communion is divided on the issue of homosexuality. The American Episcopal Church and the Anglican Church of Canada have both permitted gay marriage, while more conservative African and Asian branches remain opposed to it.
Premier's contacted the Church of England for a response to the claims.
