Health and Social Care Committee
The health minister conceded that the government may reconsider the role of relatives in consenting to a deceased's person's organs being used for transplantation on 20 February 2013.
Lesley Griffiths was giving evidence to the Health and Social Care Committee.
The committee is considering the Human Transplantation (Wales) Bill.
A proposed opt-out system would mean everyone is deemed to be a willing organ donor when they die, unless they state otherwise.
The bill says if a potential donor has not opted out, their next of kin can tell doctors about the wishes of the deceased.
The minister faced questions from AMs about how family members and friends are involved in decisions about taking someone's organs after death.
The Welsh government is proposing to create an unranked list of relatives and friends who would have the right to be involved in decisions about whether to take organs.
At the moment, the current system gives more weight to close relatives.
Health Minister Lesley Griffiths said she would "reflect" on potential problems with an unranked list.
Conservative AM Darren Millar said a ranked list would make it clear "who has the final say" about whether someone's organs can be taken if they have not opted out of the donor register.
Labour AM Mick Antoniw said: "It's the lack of transparency that concerns me and that's the one thing that keeps coming back from some of the witnesses we have had.
"One of the experts that came to give us evidence actually said if it is the case that any member of the family can object then that will actually probably lead to fewer donations that we actually have at the moment."
AMs were told that when someone has not opted out, human rights legislation means there must be a safeguard so families and friends can provide information about whether or not the deceased consented to their organs being taken.
However, Mrs Griffiths said: "Clearly the unranked list is a cause of concern for some people I think perhaps we do need to reflect on this."
Ms Griffiths also claimed that £2.9m would be sufficient funding for a ten year awareness campaign educating the people of Wales about the change in the law.
Liberal Democrat leader Kirsty Williams was concerned that after the initial spend for the first two years following implementation of the bill, only £50,000 per year would be spent on raising awareness.
Government officials said that the campaign would have to ensure the public were well informed as "consent couldn't be valid if people didn't understand the system."
Phil Walton, who is the team manager for south Wales specialist nurses for organ donation also gave evidence to the committee.