Judges are considering whether the man at the centre of the mystery celebrity threesome injunction can go to the Supreme Court to stop his and his husband's names being published.
The Supreme Court received an application on Tuesday from the man, identified only as 'PJS', for permission to appeal to the country's highest court.
The press in England and Wales cannot name him and his world-famous spouse YMA but the pair's identity has been reported by news organisations in the US, Scotland, and Australia.
On Monday, the Court of Appeal said the injunction banning the press from naming the celebrity couple fighting to have allegations of an extra-marital threesome from being reported should be lifted.
That decision meant that the couple had until 10am on Tuesday to formally launched an appeal against the court's ruling.
Their anonymity remains intact until judges at the Supreme Court make a decision.
A Supreme Court spokeswoman said on Tuesday that Supreme Court justices were considering his application.
The spokeswoman said the Supreme Court had received a written application from the man seeking permission to appeal.
She said justices were considering the application and added: "A decision whether to grant, or refuse, permission is expected to be made by the end of the day."
The pair had fought to ban the Sun on Sunday from running a story exposing the alleged affair to protect their children’s right to privacy.
Having obtained the injunction to protect their children’s privacy, the Court of Appeal ruled the injunction could be lifted, after The Sun On Sunday returned to court to challenge it.
“Much of the harm which the injunction was intended to prevent has already occurred,” said Lord Justice Jackson in Monday’s ruling.
“The court should not make orders which are ineffective.”
The case has reignited the thorny debate over what constitutes “in the public interest” with newspapers passionately arguing that British law is not keeping up with “the age of the Internet”.
During the hearing on Friday, Lord Justice Jackson said the identities of those involved was available to “anyone” as he suggested the claimant’s fears about the impact on their offspring could no longer stand.
Considering the appeal he accepted that “sooner or later” the youngsters “will learn about this” and added: “Those who are interested already know.”
Such injunctions have been described as “increasingly unsustainable” when the internet empowers people to publish the names themselves or simply look up who is who.