He has confirmed his legal team was analysing "information" involving McCullum. A photograph of Cairns - taken by the Herald on Friday shortly after the Herald website revealed he was to face a charge of perjury - shows him carrying a clear plastic folder labelled "BMC".
"I hope that, through this forum, significant additional information will be flushed out that will help people to better assess the situation, as well as the character and motives of the parties involved," Cairns said in a statement on Friday.
The interim injunction to stop publication was granted after McCullum swore an affidavit about his former professional relationship with mental skills coach Kerry Schwalger.
Garth Gallaway, McCullum's lawyer, said a court date for the injunction hearing is yet to be set but declined to comment further.
Mr Schwalger worked with McCullum until earlier this year.
Karyn Arkell, a publicist for Cairns, said she would refer the Herald query to him.
McCullum is likely to be a key witness against Cairns if the perjury charge proceeds to trial in London.
The Black Caps captain has given a statement to the anti-corruption unit of the International Cricket Council where he claimed to be approached by a former top player, dubbed "Player X", who asked him to fix matches.
Cairns has conceded that McCullum was referring to him, but has repeatedly denied match-fixing and described the allegations as "despicable lies".
The perjury charge stems from Cairns' successful defamation case in 2012 where he won substantial damages against Lalit Modi, the former boss of the Indian Premier League, over allegations of match-fixing.