The Duke of Sussex and other high-profile claimants who lost their High Court privacy case against the publisher of the Daily Mail face a potential legal bill of millions of pounds.

The group – which includes Sir Elton John, Baroness Doreen Lawrence and Elizabeth Hurley – sued Associated Newspapers Limited (ANL) over claims of unlawful information gathering.

But earlier this month, all their claims were dismissed by Judge Mr Justice Nicklin following an 11-week trial.

At the High Court on Wednesday, the parties in the case began a two-day hearing to decide who will pay what legal costs for the mammoth trial.

In written submissions, lawyers for the Duke of Sussex and the other claimants described the costs incurred by the Daily Mail’s publisher – almost £34.5m as of 9 July – as “eye watering”.

They claimed that ANL had exceeded its approved costs by over £18.6m, and added the group was insured to cover some of the publisher’s costs.

Nicholas Bacon KC for the claimants told the court the group took out insurance cover in the sum of £16.2m , and that ANL had “overspent what they’d told us they’d actually spent”.

The total bill the claimants could face has not been decided.

But the “current level of cover would be insufficient to meet their total costs exposure, possibly to the tune of many millions of pounds.”, the court was told.

ANL has argued the claimants should pay back full costs incurred by the publisher and has applied for the costs to be paid on an indemnity basis – a more favourable financial payout.

The publisher has argued the claimant’s side acted unreasonably during the case – a conduct clause which could trigger indemnity cover.

In written submissions, lawyers for ANL said: “Where claims involve wide-ranging allegations of dishonesty and serious criminal conduct against named individuals, a focused, responsible and proper approach is required as the norm.

“The Claimants’ conduct of this litigation wholly disregarded these key disciplines. In many cases this was in the face of repeated warnings by the Court as to the proper approach to be taken.”

Lawyers for the claimants said they conducted themselves “honestly and in good faith” and argued that an outcome where the claimant’s had to pay full costs would be “unjustified”, and added that it would have a “particularly harsh impact on the claimants who will bear personal liability” for cover in excess of their insurance.

“They have approached this litigation in good faith, under the burden of decades of intense, intrusive media scrutiny into their private lives.”

The hearing continues.