The consultation runs until 7 December and details can be found on the HFEA website.
The HFEA has advised the government to legalise the treatment, but only in licensed clinics.
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Prof Lisa Jardine, the Chair of the HFEA said she expected some strongly-held views on the subject.
But she said more evidence was needed before the HFEA recommended a move to single embryo transfer.
The conclusions of the three month consultation will be presented at the HFEA board meeting in July.
Patients will be able to access the HFEA's advice on potential risks on its website from next month.
Guidance from the Human Fertilisation and Embryology Authority (HFEA) is to be updated in the light of US research.
The HFEA has said that the primary reason it allows "embryo selection" is to avoid serious disease in the resulting child.
But a HFEA spokeswoman said discussions with focus groups held before the consultation process began did not indicate that would be a problem.
James Healy of the HFEA said it was aware researchers at Newcastle and Edinburgh universities were keen to be able to obtain donated eggs.
Trish Davies, director of regulation at the Human Fertility and Embryology Authority (HFEA), backed the strategy and said they would be monitoring clinics' performance.
The HFEA is unable to offer advice on clinics outside the UK but says couples need to consider the risks and implications of going abroad for treatment.
In the UK the HFEA (the Human Fertilisation and Embryology Authority) concluded that "sufficient arrangements are in place to protect the welfare of the child in these circumstances".
"Following the publication of a US study into birth defects, HFEA's scientific and clinical advances committee reviewed our guidance and advice about the risks of treatment, " he said.
Juliet Tizzard, head of policy at the HFEA, said the current rules did not work and "some donors are out of pocket and they do feel undervalued at times".
Dame Suzi Leather, chairwoman of the HFEA, said they had received some complaints about treatment that might be dangerous or banned in the UK but had no powers to act.
Mori is to carry out a survey of 2, 000 people as part of the consultation process, but the HFEA said it was unlikely to ask if parents would use sex selection themselves.
The HFEA also wants any national strategy to cover not just the fertility treatments which it currently regulates, but also treatments such as donor insemination and fertility hormone injections which also contribute to twin and triplet births.
To guard against these kind of concerns, the HFEA suggests friends and family of scientists would be permitted to donate eggs, but would be given independent counselling to ensure they were acting entirely voluntarily and have not been pressured into it.
The body that regulates IVF clinics in the UK, the Human Fertilisation Embryo Authority (HFEA), concedes that provision is "patchy" in parts of the country but says both sperm and egg donation was falling long before the change in the law.
Juliet Tizzard, policy director of the Human Fertilisation and Embryology Authority (HFEA), believes this was the right thing to do, but donor numbers have declined, and many people go abroad particularly because there is still anonymity, and many donors prefer it too.
Some fertility experts - Dr Lieberman included - have queried the way in which the HFEA's patient data is presented, fearing that couples may be attracted to a particular private clinic's success rate without taking into account other factors - such as the age of its patients or the techniques used.
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