Same-sex marriages are now banned in the Anglican church in both England and Wales, although Scotland voted to introduce the ceremonies earlier this year.
The general synod, the body which scrutinises and approves legislation within the church, will now debate the option of holding official services for couples who have taken part in secular marriages or civil partnerships.
The OneBodyOneFaith group, which campaigns for LGBT integration with the church, expressed support for the move.
The Bishop of Hereford, the Right Reverend Richard Frith, said: "Clergy are already encouraged to respond pastorally and sensitively when approached".
It stops short of calling for same-sex weddings to be conducted in church, but calls for priests to be allowed to offer a formal blessing in church to same-sex couples who have had a civil wedding.
The general synod will now debate a form of service described as "neither contrary to nor a departure from" the doctrine of the church.
A gay member of the synod, Jayne Ozanne, said she was thrilled with the support for the Hereford proposal.
However, Suzie Leafe, leader of conservative group Reform, slammed the move.
The motion, which was put forward following requests from same-sex couples for a blessing service, suggests parishes should be free to opt out.
"Marriage [is] a lifelong relationship between a man and a woman, and this has been, and still, is the understanding of the vast majority of the worldwide church for two millennia", she said.
"It is recognised, however, that there is real and profound disagreement in the Church of England over questions relating to human sexuality and the House of Bishops has recently embarked on the preparation of a major new teaching document on marriage and sexuality".