According to reports in the Guardian, the paper will “defend robustly a legal action seeking to force the disclosure of the documents that formed the basis of its Paradise Papers investigation”.
This was echoed by the BBC, which added that the revelations were in the “highest public interest”.
The law firm said it was “obliged to take legal action” as the information had been taken in a “criminal act”.
The identity of the hacker(s) has not yet been uncovered.
In legal correspondence, Appleby has also demanded that the Guardian and the BBC disclose any of the six million Appleby documents that informed their reporting for a project that provoked worldwide anger and debate over the tax dodges used by individuals and multinational companies, the paper reported.
Appleby is also seeking damages for the disclosure of what it says are confidential legal documents. It claims that the documents were stolen in a cyber-hack and there was no public interest in the stories published about it and its clients.
It has only brought legal action against the Guardian and the BBC, the newspaper reported.
Both organisations have said they will strongly defend their roles and conduct in reporting the Paradise Papers.
The BBC said its “serious and responsible journalism” had revealed matters which would otherwise have remained secret and that authorities around the world were taking action as a consequence.
The Guardian said the legal action was an attempt to “undermine responsible public interest journalism”.
The 13.4 million Appleby documents were leaked to the German newspaper Süddeutsche Zeitung, which shared them with a US-based organisation, the International Consortium of Investigative Journalists (ICIJ).
The leak from Bermuda-based Appleby was the second law firm hack to hit the offshore world. It followed the Panama Papers in April 2016, which saw 11.5 million documents exposing the offshore dealings of politicians, sports stars and public officials.
That data was stolen from Panamanian law firm Mossack Fonseca.