The owners of Ashers bakery are seeking clarity on their legal position in regard to a potential Supreme Court challenge. The McArthur family are also considering taking their case to the European Court of Human Rights if the route to the doors of the Supreme Court is not open.
Last month, Court of Appeal judges in Belfast upheld an original judgement which ruled that Ashers had discriminated against a gay activist for declining his order for a cake bearing the slogan "Support Gay Marriage".
The McArthur family's legal team believe an appeal to the Supreme Court may not be possible under legislation relating to appeals in civil cases.
The lawyers have written to the Appeal Court judges in Belfast requesting that they issue a ruling making clear the position.
The letter stated: "In view of the complexity of these issues... and the wider public importance which this case clearly has, and in order to make clear that the appellants (Ashers) have exhausted their domestic remedies... we respectfully invite the Court of Appeal to consider giving a short ruling on the question of whether appeal to the United Kingdom Supreme Court is available in this case".
Simon Calvert, from The Christian Institute, which has backed the McArthur family throughout the legal proceedings, 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."
If the Supreme Court is not an option, the Ashers could still mount a challenge in the European courts.
The Christian Institute said Northern Ireland's Attorney General John Larkin could also potentially refer "devolution issues" linked to the cake case to the Supreme Court.
Northern Ireland-based Ashers, a name with Biblical connotations, has six branches, employs more than 80 people and delivers across the UK and Ireland.
Gareth Lee, a member of the LGBT advocacy group Queer Space, had wanted a cake featuring Sesame Street puppets Bert and Ernie for a private function marking International Day Against Homophobia.
He placed the order in person at Ashers' Belfast city centre branch in May 2014. It was accepted and he paid in full but, two days later, the company called to say it could not proceed with the cake due to the message requested.
In the original court case, District Judge Isobel Brownlie ruled that religious beliefs could not dictate the law and ordered the firm to pay damages of £500.
Mounting an appeal, Ashers contended that it never had an issue with Mr Lee's sexuality, rather the message he was seeking to put on the cake.
The business said the slogan was inconsistent with their deeply held religious beliefs.
The appeal was heard before three senior judges at Belfast's
Court of Appeal in May, with the reserved judgment delivered in October.
In delivering their ruling, Northern Ireland's Lord Chief Justice, Sir Declan Morgan, said Ashers had directly discriminated.
He rejected the argument that the bakery would be endorsing the slogan by baking the cake.
"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," he said.
Simon Calvert from the Christian Institute added: "They're fine, I've just been speaking with Daniel McArthur. When they talk about trusting in God, they mean it. They do have a remarkable peace about them, as a result of that.
"He wrote to say how thankful he was for their prayful support. He said it was a joy to take a small stand for Christ and to have opportunities to witness for him in the public spotlight.
"They don't feel defeated or dismayed and they think that God will use things to bring glory to himself."
Click here to hear Simon Calvert speaking with Premier's Alex Williams: