The Christian bakery at the heart of the "gay cake" case wants to take their case to the UK's highest court.
Ashers Baking Co. were found to have discriminated against an LGBT customer by refusing to bake a cake with the slogan "support gay marriage". The initial ruling by a County Court was upheld by Northern Ireland's Court of Appeal in October but now the family-run company want to take their case to the Supreme Court in the London.
The legal team behind the Christian family believe the route to the UK's top court may be blocked but have written to Northern Ireland's judges for clarification.
If that option is not available, the family will consider taking the issue to the European Court of Human Rights (ECHR) in Strasbourg.
Simon Calvert, spokesman for the Christian Institute who are supporting Ashers, said: "Under the complex rules regarding appeals in civil cases, such as the Ashers case, the Court of Appeal decision seems to be final, according to the terms of the Judicature Act 1978."
But because of "wider public importance" and "complexity of these issues", the lawyers asked the Court of Appeal "to consider giving a short ruling on the question of whether appeal to the United Kingdom Supreme Court is available".
The appeal comes after judges in Belfast ruled Ashers had discriminated against customer Gareth Lee, an LGBT campaigner, for declining his order based on their Christian beliefs.
Ashers argued their refusal was not due to Lee's sexual orientation but because of the message on the cake - "support gay marriage" - which contradicted their religious beliefs.
But the court found "this was direct discrimination" because they would not have objected to a slogan that supported heterosexual marriage.
Lord Chief Justice Sir Declan Morgan added: "The fact that a baker provides a cake for a particular team or portrays witches on a Halloween cake does not indicate any support for either."
After the judgement Michael Wardlow from the Equality Commission of Northern Ireland (ECNI) said the "freedom to express religious or political beliefs have to be balanced with laws to protect the most vulnerable".
He added when "people of faith enter commerce they should decide how they limit what they offer".
Daniel McArthur, general manager of Ashers, said he was "disappointed" with the verdict. "This ruling undermines democratic freedom. It undermines religious freedom. It undermines free speech.
"We had served Mr Lee before and would be happy to serve him again. The judges accepted that we did not know Mr Lee was gay and that was not the reason we declined the order. We have always said it was never about the customer, it was about the message. The court accepted that. But now we are being told we have to promote the message even though it's against our conscience."
The family have not confirmed whether they will take the matter to the ECHR if the Supreme Court is not an option.