Blogger Murray was sentenced to eight months in jail at the High Court in Edinburgh last month, relating to material capable of identifying four complainers in his coverage of the Alex Salmond trial.
The former ambassador to Uzbekistan watched two days of Mr Salmond’s trial in March last year from the public gallery of the court, writing about the case on his website.
Judges subsequently ruled that he was in contempt of court due to the risk of jigsaw identification.
Following Lady Dorrian’s sentencing in May, he was given a three-week period to appeal the decision.
During a hearing at the High Court on Monday, his lawyer Roddy Dunlop QC argued Murray should be granted permission to appeal to the UK’s highest court against the verdict of contempt of court and the sentence.
The notion of jigsaw identification had received “relatively little attention” from the courts, he said.
He said: “His submission is that imprisonment in a case such as this would be compliant with Article 10 (of the European Convention of Human Rights) only in exceptional circumstances. My submission is that that is not the case here.”
He continued: “The proliferation of blogging makes the outcome here one of general importance.
“This is an issue of acute sensitivity and one on which the view of the Supreme Court would be very welcome.”
Mr Dunlop said some had suggested the proceedings against Murray were “politically motivated”.
The QC said he did not make this submission himself, but “the fact remains these views are out there”.
Mr Dunlop also argued that if the court were to find against his client, then Murray could still appeal directly to the Supreme Court.
If that were the case, he argued that the arrest warrant should be suspended for a further three or four weeks.
Lord Justice Clerk Lady Dorrian said the court would consider the submissions it had received and hoped to issue a written decision on Tuesday.