Actress Sienna Miller and former footballer Paul Gascoigne have agreed a settlement with News Group Newspapers (NGN) over alleged phone hacking.
However, they are asking a High Court judge to rule in their favour in a dispute over the wording of a statement they wish to be read to the court.
The pair are among a number of high-profile figures to have brought legal action against the publisher of the News Of The World and The Sun.
They alleged their privacy was invaded.
Ms Miller and Mr Gascoigne have each agreed a settlement with the publisher, but NGN took issue with the wording of a statement they wanted to be read to the court on their behalf. The company is asking Mr Justice Fancourt to resolve the dispute.
The judge is also being asked to decide what words should be included in a statement on behalf of former Liberal Democrat MP and campaigner Dr Evan Harris.
Barrister David Sherborne, who is representing the claimants, told the court the matter was of great importance to them, as demonstrated by the fact Ms Miller had flown to the UK from the US to attend the hearing.
Ms Miller is expected to give a statement outside court later on Wednesday, once the issue has been resolved, as is Gerald Shamash, solicitor for Mr Gascoigne.
Mr Justice Fancourt was asked by NGN lawyers to conduct parts of the hearing in private, but he refused, saying that the allegations were already in the public domain.
Anthony Hudson QC, representing NGN, said the three claims linked to the statements under dispute were "explicitly settled on the basis of no admission of liability".
He said allegations in the statements were "in reality being presented as fact and that is inappropriate", adding that this "runs completely contrary to the basis on which the claimants settled".
Mr Sherborne told the court Ms Miller was "perfectly entitled" to include certain details in her statement, pointing to case law on unilateral statements in open court which he said made it "very clear" that she should be "allowed to say what she felt was hurtful to her, damaging, the effect of the publication and what she wants to say about the settlement".
The court earlier heard statements from a number of celebrities and high-profile figures who have settled phone hacking claims against the publisher of the News Of The World.
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Statements were read on behalf of 15 celebrities and other figures, including actor Sean Bean, Texas lead singer Sharleen Spiteri and ex-cricketer and commentator Shane Warne.
News Group Newspapers (NGN), publisher of the now-defunct newspaper, has agreed to pay confidential damages to each of the claimants and also pay their legal costs.
The publisher, through its legal team, made public apologies for the actions of the News Of The World, but did not admit any liability in relation to allegations of phone hacking at one of its other newspapers, The Sun.
Other celebrities to have settled cases include actresses Julia and Nadia Sawalha and Michelle Collins, ex-television presenter Dani Behr, singer Dane Bowers, and former Coronation Street actor Richard Fleeshman.