New Zealand Cricket (NZC) chief executive Justin Vaughan says it is regrettable that John Howard was ever nominated for the ICC vice-presidency.
The former Australian prime minister's candidacy was rejected by cricket's governing body last month, although Cricket Australia (CA) are yet to rule out re-nominating him.
Six of the 10 Test-playing nations opposed Howard's joint nomination from NZC and CA.
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The search for a replacement nominee was further complicated when NZC's original preferred candidate, Sir John Anderson, last week declined to reapply for the role.
NZC and CA now have until August 31 to put forward a new name for the position, which will see a successful candidate graduate to the presidency in 2012.
"In perfect hindsight, if we'd known what the ICC's position on John Howard was going to be - it is regrettable," Vaughan said.
"We felt clear messages coming back around John Howard. Australia invested a lot of time into getting Howard to put himself forward for the position, so it's a sensitive position for them.
"It's something that we'll discuss with them in the next couple of weeks."
NZC met on Friday to discuss the matter, following which their former chief executive Chris Doig and current chairman Alan Isaac emerged as leading contenders.
"Both those people certainly tick the boxes," Vaughan said.
But a CA spokesman said Australian cricket were yet to decide who they wanted in the role and refused to rule out pushing to retain Howard as their candidate.
"We're not going to go into hypotheticals ... he's the only name our board has authorised at this point," the spokesman said.
He said CA chairman Jack Clarke arrived back in Australia on Monday, having been overseas since the ICC board meeting in Singapore in June.
Clarke and Isaac were expected to begin discussions in early August.
But Vaughan said NZC wanted a quicker resolution than the long-running process that originally resulted in Howard being nominated over Anderson.
And he said the ICC had made their position clear on Howard.
"It's really important NZC and CA are on the same page with the new nominee ... they (the ICC) said (Howard) was unacceptable and asked us to re-nominate. That's pretty clear."
