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The Government has confirmed this is an “important issue” and that it will consider changes to the law.
Norway has banned marriage between close relatives. Officials in Sweden last month recommended vetoing cousin marriages, and similar measures are expected in Denmark.
The move is fuelled by concern about the higher risk of birth defects and the potential for women to be coerced into these unions.
In the US, lawmakers in Tennessee this year voted to bring in a ban, while in the UK, former Tory Party chairman Richard Holden is the latest high-profile figure to urge the Government to look at restricting unions between first cousins.
He was at the forefront of successful efforts to ban “virginity testing” and a surgical procedure designed to give the impression that women are virgins. He now wants the Government to focus on this issue.
The MP said: “There are serious concerns regarding first cousin marriage. It leads to around twice the level of birth defects versus the general population and can be used to re-enforce negative structures and control women.”
But Sarah Norcross, of the Progress Educational Trust, said it would be “misguided to seek to make a criminal offence of sexual activity between consenting adults who happen to be first cousins”.
Ms Norcross acknowledged “people whose parents are first cousins are at a higher risk of certain genetic conditions” but said the majority of children “will be healthy”.
Justice minister Alex Davies-Jones said: “We are aware that all aspects of weddings, including first-cousin marriage, are important issues.” Restrictions would require changing the 1949 Marriage Act and the 2003 Sexual Offences Act.
Ms Davies-Jones said the Government would “take time to consider our marriage law” before “setting out our position”.