Fairfax said it was the only statement Cairns was prepared to release to the media before conferring with lawyers.
Meanwhile, Vincent confirmed he is cooperating with the ICC investigation.
"I wish to let everyone know that I am cooperating with an ongoing ICC Anti-Corruption investigation that has been made public today."
"This investigation is bound by a number of rules and regulations that mean I am unable to make any further public comment."
"I will personally talk to the public when I am able to. In the meantime I cannot comment. Please respect me and my family's privacy until such time."
Cairns is working for Sky TV as a commentator in Dunedin at the moment. Tuffey is believed to be living with his partner in Sydney while Vincent's whereabouts is unknown at the moment.
Sky TV spokesperson Kirsty Way said Cairns left the ground shortly after he was named.
"This was news to Sky and we understand it was news to Chris," she said.
"He has elected to stop commentating on the current test match and come back to Auckland to be with his family. Sky will be talking to him over the coming days."
When asked to comment on the allegations by an Otago Daily Times reporter, Cairns declined to comment.
No charges have been laid.
The news follows the revelations in today's Herald about the lengthy investigation concerning matches in more than one country.
The allegations continue to make headlines around the world - with particular interest from cricket-mad countries like the UK and Australia.
The UK's Guardian and Daily Mail news websites named Cairns and Vincent as being under investigation by the ICC.
The BBC and The Mirror were also reporting on the allegations but had not named any players.
In Australia, news outlets had also picked up on the investigation.
National broadcaster ABC, and daily newspapers in Sydney and Melbourne continued to cover the story. Melbourne's The Age newspaper, as well as the Sydney Morning Herald and The Daily Telegraph featured headlines referencing a match-fixing investigation involving New Zealand cricketers
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- with additional reporting by Teuila Fuatai of APNZ