Harry, and former Labour deputy leader Tom Watson, are bringing legal action against News Group Newspapers (NGN) over allegations of unlawful information gathering by journalists and private investigators between 1996 to 2011.
NGN, which also ran the now-defunct News Of The World, denies any unlawful activity took place at The Sun.
Rupert Murdoch's newspaper group has settled more than 1300 lawsuits involving celebrities. (AP PHOTO)
The trial was due get underway on Tuesday but delays mean that the case remains unopened, amid speculation over whether Harry and Watson have settled their claims.
After two earlier requests for adjournments on Tuesday, thought to be related to settlement discussions, Justice Fancourt refused a third request for a delay as both sides had had "ample time to seek to resolve their differences".
Following a short break, lawyers for both sides asked for the green light to challenge the judge's decision to not provide a further delay.
The case involving allegations of unlawful information gets underway in a London court. (AP PHOTO)
NGN's barrister Anthony Hudson said the parties "have been involved in very intense negotiations over the last few days and the reality is we are very close", adding that there were "time difference difficulties".
"Very unusually, both parties are in complete agreement that this is a very important step.
"The number of times the parties have been in agreement in this litigation are very few and far between."
The more than 1300 lawsuits settled by NGN include cases involving celebrities, politicians, well-known sports figures and ordinary people who were connected to them or major events.
Justice Fancourt said he would not allow both sides to go to the Court of Appeal himself over his decision, but that he would not "stand in the way of access to justice" if they wanted to ask appeal judges for the green light.
The hearing will resume on Wednesday at 10am (9pm AEDT).
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