Currently, anyone who wants to become an organ donor when they die must register.
Under new proposals put forward by Health Minister Edwin Poots, people would be presumed to have given consent for their organs to be donated upon their death, unless they have opted out.
Peter Robinson and Martin McGuinness gave the opt-out campaign their backing at the Belfast City Hospital on Tuesday, where they toured the Regional Nephrology and Transplant Service with former GAA player Joe Brolly.
Mr Brolly became a high profile campaigner for organ donation after he gave a kidney to a friend last year. The transplant was unsuccessful after medical complications.
The former All-Ireland footballer told UTV he has no doubt that an opt-out system would be a massive success in Northern Ireland.
Under the proposed system, the family will always be asked for final consent, and Mr Brolly said his campaign is raising awareness of the need for increased donation.
There are currently around 200 people on the active transplant waiting list for Northern Ireland, while 15 die each year while waiting.
It's a gift from one family to another. It's a sacred gift at a very difficult time.
"It's going to save lives. Up until now this was an invisible tragedy," Mr Brolly added.
"People will start thinking about organ donation and the difference it makes. They will see the massive difference that soft opt-out makes because the statistics can't be argued against," he explained.
The living donor said he hopes organ donation will become the norm for society, as he has seen the difference it can make to people's lives.
"A family who I spoke to recently allowed their son's organs to be used. He saved seven lives. Seven different people benefited from his organs. So that gives you some idea of the effect that this can have on the community. This is a brilliant thing.
"What's the downside?" he asked.
Mr Robinson, who described Northern Ireland as a very caring society, said it was a worthwhile experience to talk to an altruistic, live donor.
"If you can put yourself in the place of a mother or father at the side of the bed of someone who's going to die for the want of an organ, or whose quality of life is going to be dramatically worse for the want of an organ, I think you will understand the imperative that there is for the campaign," he said.
Mr McGuinness also paid tribute to Mr Brolly for what he called his "huge personal sacrifice to highlight the important issue of organ donation".
"Joe's awareness raising campaign has set out the benefits of a presumed consent option. The top ten organ donor rate countries in the world all have an opt-out system," he said.
"I support the opt-out system as the best way forward to increase organ donations."
Public support would be critical for the success of opt-out legislation, but just having the debate on opt-out should serve to assist in raising the profile of organ donation and increasing the number of possible donors.
Mr Poots said he is looking for feedback on the proposals and they could lead to a change in legislation. He added that he has personal reasons for backing the move.
"My mother might have been alive had she had the opportunity to have a liver transplant whenever she was younger," he said.
"I think that there's a great opportunity out there to ensure that as many people as possible have the opportunity to live a full and healthy life, who otherwise would not," he said.
SDLP Health Spokesperson Conall McDevitt has welcomed the announcement.
"An opt-out scheme for organ donation would be a huge step forward for this region, and would bring hope to so many patients and their families currently on the waiting list for an organ donation," the south Belfast MLA said.
"Research from other organisations has shown high levels of support for this sort of scheme, and I welcome the Minister's recognition of this fact."
Sinn Féin Health Spokesperson Sue Ramsey also welcomed the consultation.
"It is important that we maximise the number of organs that are available for transplant as for every one person who receives a transplant three other die due to a lack of organs available," the west Belfast MLA said.
"Under the present system only 25% of the population carry a donor card yet in every survey carried out around 85% of those asked support organ donation.
"I would encourage people to get on the donor list and let their families know their feelings in terms of donation so that many more lives can be saved."