At a meeting earlier this week eight bishops were in favour of allowing religious weddings for same-sex couples, against four of the opposite opinion.
However, aware that the question divides the Church, the majority of the bishops advocate a simple blessing of civil unions within a church setting.
But that scaled-down option also meets opposition from a minority of bishops.
The question has to be agreed upon at the next synod in April 2014.
Norway is one of the most liberal countries in Europe when it comes to homosexuality.
Same-sex civil marriage and adoptions have been legal since 2009, and the Church of Norway also allows the ordination of homosexuals.
Neighbouring Sweden authorised religious same-sex marriage in 2009.