MáirtÃn Ó Muilleoir said in a letter responding to another Member of the Local Assembly (MLA) that he wants the legislation passed as quickly as possible.
Northern Ireland is the only Home Nation where same-sex marriage is not legal.
Bills legalising same-sex marriage have been blocked before by a coalition of MLAs led by the DUP, which formally opposes same-sex marriage.
Catholic bishops in Northern Ireland have said before previous failed votes on same-sex marriage: "Those who vote in favour of this motion have no way of knowing what the full consequences of such a vote will be.
"The truth about marriage derives from its intrinsic nature as a relationship based on the complementarity of a man and woman and the unique capacity of this relationship alone to generate new life.
"This truth does not change with the shifting tides of historical custom or popular opinion."
Same-sex marriage became legal in England, Wales and Scotland in 2013.
It became legal in the Republic of Ireland in 2015 after a referendum.