"Sorry, I cannot say any more as it could jeopardise further investigations," White said.
The press conference was called to respond to a story in London's Daily Telegraph that suggested former Black Cap Lou Vincent was seeking a plea bargain for full disclosure of his involvement in spot- and match-fixing.
McCullum is playing in the Indian Premier League for the Chennai Super Kings. It is understood he immediately and emphatically rejected the approach. Approached by the Herald last night, he had no comment.
White did confirm that matches involving the Auckland Aces in South Africa at the 2012 Champions League - an international Twenty20 competition - were under anti-corruption investigation, a story broken in the Herald in December.
"No games played in New Zealand are being investigated by the ICC," White said. "No current Black Caps are being investigated. No matches involving New Zealand national teams are being investigated.
"However, we have been informed by the ICC that some Auckland Aces matches in the Champions League in South Africa in 2012 are being investigated.
"I would like to stress that we understand that this is very much an isolated incident. Match-fixing is a threat to cricket around the globe and we remain 100 per cent behind the ICC in their focus on fighting corruption."
White said NZC knew about the Aces' involvement when the story broke, but could not comment because it did not have ICC permission.
Vincent, a former New Zealand batsman, is co-operating with the anti-corruption unit and London's Metropolitan Police as they investigate claims of spot- and match-fixing in Asia, South Africa and England.
New Zealand allrounder Chris Cairns and seam bowler Daryl Tuffey are being investigated. Both deny any wrongdoing.
Cairns said: "I'm waiting to conclude the interview I started with the Met [police] over six weeks ago here in New Zealand.
"Obviously the time it's taken to conclude this is frustrating but I continue to co-operate fully with the relevant authorities."
Cairns' friend and former lawyer, Andrew Fitch-Holland, was brought back to Scotland Yard yesterday for a second interview, before being released on bail.
He was arrested in March on suspicion of perverting the course of justice, relating to evidence given at Cairns' successful London High Court defamation lawsuit against former Indian Premier League boss Lalit Modi.
The IPL impresario made allegations against Cairns on Twitter.
Fitch-Holland's lawyer, Tony Wyatt, QC, said the barrister was questioned yesterday "about a number of evidential matters which he wasn't questioned about last time".
"The material was all very much the same but it wasn't disclosed to him on the last occasion. This time it has. He answered questions about everything that had been disclosed."
Fitch-Holland is likely to be called back in August or September for another police interview.
"He will be re-interviewed; alternatively, they may decide they have enough [evidence] and charge him at that time," Mr Wyatt said.
"The other option is they may decide they're not going to charge him ...
"He's not been charged but the investigation is ongoing. It seems most of the information is coming from one source, so the decision they will make when they come to decide to charge him is whether that one source is credible."
That one source is almost certainly Vincent. He has said he is fully co-operating with police and ICC anti-corruption investigators, and the Telegraph yesterday described his evidence as a "treasure trove of information".
The Herald contacted Vincent last week via email, seeking comment on his role in the investigation.
"I still can't say anything or make comment," Vincent replied.
Cairns, Vincent and Tuffey played for the Chandigarh Lions in the short-lived Indian Cricket League.
The ICC last night refused to shed any further light, saying it didn't comment on corruption investigations.
Key dates
• December 5, 2013: Herald reveals three NZ cricketers under investigation.
• December 6: Chris Cairns' legal team strikes back.
• December 9: Cairns': I'm still in the dark.
• December 14: ICC probes Aces match in South Africa.
• Feb 10, 2014: "Who's putting this in the media?" - Cairns.
• February 27: Vincent admits approach by bookmaker.
• March 27: Cairns' lawyer arrested.
