They are asking the highest court in the land to overturn decisions of both the High Court and Court of Appeal that they are not eligible to vote on June 23 because they have lived outside the UK for more than 15 years.
London-born war veteran Harry Shindler, 95, who has lived in Italy for 35 years, and solicitor Jacquelyn MacLennan, 54, from Inverness, who has lived in Belgium since 1987, are both fighting the 15-year rule.
They want the Supreme Court justices to declare the rule, contained in Section 2 of the EU Referendum Act 2015, an unjustified restriction on their EU right to freedom of movement and an infringement of their common law right to vote.
The appeal court held that the 2015 Act does not fall within the scope of EU law at all and their challenge “falls at the first hurdle”.
Both want the right to vote against Brexit in a bid to protect their way of life.
The test case is being watched by many other expats living in other EU countries who fear a Leave vote will lead to them being deprived of EU citizenship and the rights that go with it.
Ms MacLennan, who specialises in EU competition and environmental law and is a partner in the Brussels office of a global law firm, said: “Brexit would have a huge impact on my personal and professional life.
“Excluding two million citizens like me from voting - as the Government recognises - is unjust and unfair.”
Mr Shindler said: “I am still waiting for the Government to tell us why British citizens in Europe can’t vote in this referendum.
“The Government had agreed to scrap the 15-year rule before the Referendum Bill was passed, agreeing it was arbitrary and undemocratic.”
Both Ms MacLennan and Mr Shindler are barred from the referendum franchise because, having lived abroad for 15 years, they are deemed no longer to have a “closeness of connection” to the UK to justify giving them the franchise.
Mr Shindler fails the “closeness” test, despite being born in London, fighting in the nation’s armed forces and still being a taxpayer.
The UK passport holder was in the Second World War landings at Anzio. His big fear is that a Brexit vote could lose him EU citizenship and seriously disrupt his way of life in Italy.
In 2014 he was awarded an MBE for his services to Anglo-Italian relations. He still pays taxes on his pension to HM Revenue and Customs.
Rejecting the expat challenge, the appeal judges - Lord Dyson, Master of the Rolls, sitting with Lord Justice Elias and Lady Justice King - declared the 15-year rule did not amount to an unjustified restriction on EU law rights to freedom of movement or to the common law right to vote.
Summing up the ruling, Lord Dyson said the 2015 Act did not fall within the scope of EU law at all and the challenge “falls at the first hurdle”.
The judge said the EU recognised that a decision of a member state on whether or not to withdraw from the EU was “an exercise of national sovereignty” governed by a country’s own constitutional arrangements.
Even if the 2015 Act was within the scope of EU law, said the judge, it did not amount to a restriction on free movement rights.
Parliament had the right to determine the scope of the referendum, and any common law right to vote did not take precedence over an Act of Parliament, said the judge.