Judges are considering whether a celebrity who wants to keep his name out of a tabloid newspaper story can be given permission to take his case to the Supreme Court.
The man lost the latest round of his legal battle on Monday when three Court of Appeal judges ruled that an injunction barring The Sun on Sunday from naming him should be lifted.
He now wants to take his case to the Supreme Court.
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."
Sun on Sunday editors want to publish an account of the man's alleged extra-marital activities.
But the man had argued that he had a privacy right and took legal action.
The newspaper won the first round in January when a High Court judge refused to impose an injunction barring publication.
But the man appealed - and two appeal court judges ruled in his favour.
Lord Justice Jackson and Lady Justice King imposed an injunction preventing the newspaper from identifying the man in an article.
Lawyers for News Group Newspapers, publishers of The Sun on Sunday, then asked three appeal judges to lift the ban after the man's identity emerged online.
They told Lord Justice Jackson, Lady Justice King and Lord Justice Simon at a Court of Appeal hearing in London on Friday that the ban should go because the man has been named in articles abroad - outside the legal jurisdiction of England and Wales - and his name could be found on the internet.
The man opposed the application and said the ban should stay.
But Lord Justice Jackson, Lady Justice King and Lord Justice Simon ruled in the newspaper's favour on Monday.
Appeal court judges said the injunction would remain until 1pm on Wednesday to give the man time to make an application to the Supreme Court.
Desmond Browne QC, who headed the man's legal team, told appeal judges that he wanted Supreme Court justices to examine the case.
Mr Browne suggested that the Court of Appeal decision might lead to the ''rule of the press'' rather than the ''rule of the law'' and signal the death of the ''celebrity privacy injunction''.
Gavin Millar QC, who led News Group Newspapers' legal team, had told the three appeal judges: ''The information which is the subject of the injunction has now entered the public domain, has been published widely and is freely and generally accessible in this jurisdiction.''
Mr Browne disagreed and had told judges: ''The order is having the effect of preventing intrusion and harassment.''
Lord Justice Jackson, Lady Justice King and Lord Justice Simon said the ban should go.
''Knowledge of the relevant matters is now so widespread that confidentiality has probably been lost,'' said Lord Justice Jackson. ''Much of the harm which the injunction was intended to prevent has already occurred.''
He added: ''The court should not make orders which are ineffective. It is, in my view, inappropriate (some may use a stronger term) for the court to ban people from saying that which is common knowledge.''
The three appeal judges did not identify the celebrity but referred to him as ''PJS''.
They said that he was in the entertainment business, that his spouse - named as YMA - was also well-known in the entertainment business, and that the couple had ''young'' children.