A debate on the motion is due to take place in the Assembly next week.
However the petition of concern means it cannot be passed unless it receives cross-community support.
DUP Chief Whip Peter Weir said the debate "will only result in further embarrassment for those parties and individuals who avoid telling the electorate where they stand".
The first same sex civil partnership in the UK took place in Belfast in December 2005, but same sex marriage is still illegal.
Earlier this year MPs backed a bill to make it law in England and Wales, but similar proposals for same sex marriage were rejected at Stormont last year.
Mr Weir continued: "It has been made very clear that same sex marriage will not be introduced in NI and the DUP is tabling a Petition of Concern to ensure that this motion will not be carried.
"We are the only party to have a united position in opposition to the redefinition of marriage.
"However we also do not believe that a Constitutional Convention in the Republic of Ireland should have any input into policy which is wholly a matter for representatives in this part of the United Kingdom to decide."
But Sinn Féin South Down MLA Catríona Ruane argued the issue of same sex marriage was widely supported at the constitutional convention, which was comprised of political representatives from across the island and members of the public.
She said: "An overwhelming majority of those 99 people, voted for equal marriage, 81% voted that the state shall protect and provide for same sex marriage, they also voted for children's rights to be protected."
"This is an equality issue, people who are gay and lesbian, bisexual and transgender in our communities, they deserve rights.
"What the DUP need to do is go out and explain to the gay and lesbian people in their communities, why they are blocking equality. Equality threatens nobody."
Both the Presbyterian and Catholic Church have written to Assembly Members urging them to reject the motion.
But Gerry Lynch, regular churchgoer and vice chair of campaign group Changing Attitude Ireland argues that civil marriage should be open to all.
"We're talking about civil marriage here not religious marriage. It should be based on what's best for society, not on the precepts of any particular church," he said.
"As a Christian, I believe marriage is a bedrock, a foundation of society and there are already thousands of couples in Northern Ireland living in same sex relationships long term."