It was the first time wrongdoing at The Sun was acknowledged as the papers admitted "phone hacking, surveillance and misuse of private information by journalists and private investigators".
News Group attorney Anthony Hudson said there had been productive discussions and said there was a "very substantial sum" on the line if the trial began before they could complete "very intense negotiations".
Their appearance is a further sign the royal family is slowly returning to normal at the end of a year in which two of the most popular royals were sidelined by cancer.
Charles stopped short of mentioning financial reparations that some leaders at the event have urged and instead exhorted them to find the "right language" and an understanding of history "to guide us towards making the right choices in future where inequality exists".
More than 20 female ex-Harrods employees have accused Mohamed Al Fayed, whose son died along with Princess Diana in a high-speed car crash in Paris, of sexual assault.
The 73-year-old daughter of Queen Elizabeth II was walking within her Gatcombe Park estate, about 170 kilometres west of London, when an incident occurred.