Academics Rebecca Steinfeld, 37, and Charles Keidan, 41, who live in Hammersmith, west London, want a legal union which does not carry “patriarchal baggage”.
However, they are prevented from doing so because the Civil Partnership Act 2004 says only same-sex couples are eligible to become civil partners.
The couple, who have two daughters aged two and eight months, claim the government’s position is “incompatible with equality law”.
They suffered a defeat at the Court of Appeal in February last year, but were given the go-ahead in August for a Supreme Court hearing which is being held today.
Ms Steinfeld said: “We hope the Supreme Court will deliver a judgment that will finally provide access to civil partnerships for thousands of families across the country.”
Mr Keidan said: “The incredible support from many thousands of people who have signed our petition and backing from MPs across the political spectrum has enabled us to come this far.
“What started out as a personal effort to become civil partners has taken on wider significance as we realised that as many as 3.3 million co-habiting couples are affected by the status quo.
“Over the last few years, we’ve heard the same message: whilst most couples want financial and legal protection for themselves and their families, not all feel comfortable with marriage.
“Civil partnerships offer a legally binding arrangement that is fair, popular and good for families and children.”
Campaigners say it is unfair that same-sex couples can enter into a civil partnership or get married, when mixed-sex couples can only get married.
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They say this denies mixed-sex couples who do not want to get married the legal protections afforded to those in civil partnerships, such as the ability to inherit property tax-free from their partner.
They want the Government to extend the legal arrangement to heterosexual couples.