People are being asked to have their say on the ethics of allowing the creation of a baby who has three genetic parents.
The IVF procedure would be used to prevent a special category of diseases caused by inherited genes.
The consultation could pave the way to a landmark change in the law as early as next year.
Under the procedure, children could be conceived with the help of a third parent - a woman whose donated egg provides a source of healthy DNA.
A baby created in this way would have DNA from its mother and father, plus a tiny amount of donated mitochondrial DNA, which comes from the structures within cells that convert energy from food into a form that cells can use.
It is this "healthy" mitochondrial DNA that would replace the damaged DNA of its parents.
Defects in mitochondrial DNA can give rise to a range of serious and potentially life-threatening diseases including a form of muscular dystrophy.
They can also lead to the loss of hearing and vision, heart problems and intestinal disorders.
The law currently bans genetic material from being tampered with in this way.
However, ministers can make the technique legal if they are satisfied it is ethically acceptable to the public.
The final say on whether treatments can go ahead will lie with the Human Fertilisation and Embryology Authority (HFEA) which regulates IVF clinics and fertility research.
Professor Lisa Jardine, who chairs the authority, said: "We find ourselves in uncharted territory, balancing the desire to help families have healthy children with the possible impact on the children themselves and wider society."
She recalled public attitudes to the birth of the first IVF baby, Louise Brown, in 1978.
"A lot of people thought she was going to be a monster, and the idea of what would happen if you allowed conception outside the womb was seen as absolutely appalling. Those uncharted waters are now charted waters."
Critics fear that allowing mitochondrial DNA to be altered could be the start of a slippery slope leading to a "brave new world" of genetically modified designer babies.
One in 200 children born each year in the UK have some form of mitochondrial disease. Not all suffer serious symptoms.
The consultation, which aims to get the views from ordinary people rather than experts, runs until December 7, with a report being submitted to the Government next spring.
People can air their views at: www.hfea.gov.uk .