The newspaper group was seeking to lift part of the wide-ranging injunction which prevents it from reporting the position held by an RBS employee with whom Goodwin had an affair, and the duration of the affair.
Mr Justice Tugendhat reserved judgement in the case until a later date, to be fixed at another hearing.
The affair was first reported in the press after Lib Dem peer Lord Stoneham revealed in the House of Lords that it was the subject of an injunction.
News Group Newspapers, a subsidiary of News International, then went to court and successfully applied to be able to identify Goodwin in relation to the affair. Mr Justice ruled that the injunction should be varied to allow Goodwin to be named, in light of Stoneham's comments, but upheld the part of the injunction protecting the woman involved.
Lawyers representing the woman then attempted to persuade a judge to launch contempt proceedings against the Daily Mail, which they accused of deliberately flouting the injunction by naming her.
But Mr Justice Tugendhat refused to refer the matter to the attorney general, stating in court: "The reason that I decline to make the reference is that, in my judgment, it would not assist the Attorney General.
"The lady is free to refer the matter to the Attorney General herself, and the Attorney General is free to act of his own motion.
"This case has received extensive coverage in many newspapers and other news media and has been the subject of public judgments.
"If the Attorney General does decide to consider this matter, the contents of this judgment will be available to him."
Goodwin's injunction, obtained earlier this year, was first brought to light in March by John Hemming MP, who discussed its existence under parliamentary privilege in the House of Commons.
News Group Newspapers was behind three applications to the high court last month to have a separate injunction lifted that prevents the naming of footballer "CTB", who is alleged to have had an affair with model Imogen Thomas.
The second and third of those applications, all of which were unsuccessful, came on the same day that John Hemming MP named Ryan Giggs in parliament as the footballer at the centre of a privacy injunction controversy.
Both Mr Justice Eady and Mr Justice Tegendhat ruled against the newspaper group however, stating that whatever the situation with the identity of the footballer, the injunction could still protect the player from harassment.
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