The group wants to launch a public consultation on a Bill to introduce gay and lesbian marriage.
Currently a DUP veto has prevented a change in the law five times. The DUP indicated it would block this latest attempt.
Northern Ireland is the only part of the UK or Ireland to only allow marriage between a man and woman.
A number of Northern Ireland Assembly members are attempting to by-pass the administration by pursuing new legislation through Private Members' Bills.
On Monday evening, Finance Minister Mairtin O Muilleoir acknowledged those efforts but said he hoped he could still make progress through the Stormont Executive, by securing DUP agreement to launch a public consultation on a Bill from his department.
"I am in favour of marriage equality - that is why in this mandate I am determined to make what progress I can to put marriage equality on the statute book," he said during Assembly question time.
"I would like to proceed by way of an Executive Bill and I will at the earliest opportunity seek Executive agreement to consult on the issue."
He promised any legislation would include protection for clergy and church bodies not wanting to participate in same-sex marriage ceremonies.
Most churches in the region hold that marriage is between a man and a woman.
Although a slim majority of MLAs voted in favour of lifting the ban when it was debated for a fifth time last November, the proposal fell when the DUP deployed a controversial voting mechanism - a Petition of Concern - to effectively veto it.
This means that at least 40 per cent from both the Catholic and Protestant communities would need to agree on the legislation.
The DUP, which is the largest party in the Assembly, has argued that same-sex couples already have the ability to enter into civil partnerships and insisted the appetite is not there for further change.
A party spokesperson said: "The DUP supports the current definition of marriage. Our position remains as outlined during the Assembly election campaign."
