The furore started after McCullum's statement to the ICC's Anti-Corruption and Security Unit (ACSU) was leaked to the media.
In that statement, McCullum claimed that he was unsuccessfully approached for match-fixing by a top player, who has been identified merely as 'Player X', in 2008.
Speculation is rife that 'Player X' is none other than McCullum's former New Zealand teammate Chris Cairns, who has vehemently denied the charge.
"The investigations that we are talking about are at an advanced stage and should be tied up a lot sooner than that. Having said that, I've seen how when law enforcement agencies are involved it does take some time," he added.
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Reports here claim that Cairns will today leave for London to meet ICC ACSU officials and the Metropolitan Police.
Vincent, however, is an accused in the matter and has been charged on 14 counts of corruption by the England and Wales Cricket Board for allegedly fixing matches in the County Championships.
"Part of it does involve a police investigation in the UK, which I can't really vouch for as to when that will be finalised. But certainly from our perspective the investigations are close to finality and hopefully we should be making announcements pretty soon," Richardson said.