Another board member, Laurence Day, said he was not aware of any board member who wanted to see Mr Craig recalled as leader.
"How much we know about it varies...and under the circumstances I think it is prudent he stands down. I don't see, at the board level, any great enthusiasm for him to return.
"It's not that we don't respect what he's done. I mean, hey, even when he stepped down he wrote off his loan to the party. You have to respect the guy for what he's done, but obviously there are some challenges for him ahead. I can't speculate on the veracity of what is going around."
Mr Day, who received more than $10 million for his share of a private training organisation when it was sold recently, was once a National Party stalwart but now backs the Conservatives, mostly because he is a strong supporter of binding referenda.
He and his wife Katrina gave $675,000 of their own money to the Conservative Party's last election campaign.
Mr Day said he would still support the party.
"We will pick up and move on. Other parties have lost leaders, Labour has gone through a lot of blood-letting...they survive and move on. That's politics."
This afternoon, Mr Craig said he had written to Mr Stringer asking him to retract his comments, and would take further advice on a more formal approach tomorrow.
"I think once he receives that it might be wise for him to pause and take a read of that, and then maybe reconsider his position."
Mr Craig said he would only put his hand up for the leadership again if that was what a significant majority of the party membership wanted.
He had received hundreds of messages from members since stepping down, almost all of which had been supportive.
Asked if he would seek the leadership if there was support from party members but not from the board, Mr Craig said he would, provided "whatever allegations are out there are cleared up".
"If that's what the grassroots wants, I would put my name forward whether the board wanted to receive it or not. But the reality is, the board are intelligent people. If the broad membership were supportive of me, I think most of the board would have another look at it.
"The board decides on the leader within the Conservative Party. The board hasn't made clear what its process is going to be yet."
Mr Craig, the party's founder and main financial backer, stood down as leader on Friday.
He had expected to face a leadership challenge at a board meeting on the same day, which was postponed.
The board members who were planning to challenge Mr Craig's leadership were also upset about Mr Craig's appearance on TV3's late-night show Newsworthy this month, in which he was interviewed in a sauna.
They felt it was damaging to the party and reminded the public of Mr Craig's "crazier" side.
Rumours of a leadership spill first emerged in March, when some party members appeared to line up former candidate Garth McVicar for the role.
