Costs order
The claimants are also challenging a subsequent indemnity costs order. They face paying joint legal costs of up to £34.5m. Mr Justice Nicklin ordered the claimants to make an initial payment of £9.54m while the final total was calculated.
A spokesperson for the group said: "The claimants respectfully disagree with those unprecedented decisions, which they consider to be legally and factually flawed."
The statement did not reference Liberal Democrat politician Sir Simon Hughes, who was part of the action. The BBC has contacted Sir Simon to establish whether he is joining the appeal. After the costs ruling, he told the BBC he was "disappointed" and would take "the time necessary to consider both the substantive and the costs judgment and whether to pursue appeals against decisions in either judgment or both".
There was no comment from a spokesperson for Harry or Associated Newspapers.