A public statement last week, issued by the Pike River Family Reference Group, said families had reluctantly accepted a government decision to close the recovery project.
A number of families said they were blindsided by that.
The Pike River Family Committee, which is separate to the reference group, met last night and say three-quarters of the families want to self-fund a feasibility study into recovering the mine's main ventilation fan.
Last week's statement by the Pike River Family Reference Group - which is the official group working with the government's mine recovery agency - sparked uproar among some relatives of the 29 dead miners.
The reference group represents 27 of the families. It said the families accepted advice from Pike River Re-entry Minister Andrew Little that there would be no more money to expand the project.
Little told RNZ's Checkpoint programme the Government did not overcommit or under-deliver on promises.
"We made a commitment to recover the drift and do the forensic work and examination required to assist the police in their investigation. That pledge was signed up to in 2017 before the election ... that was a pledge drafted by the families.
"We spent 10 months examining and evaluating and doing all the risk assessments required to do the job and at the end of that I made a public announcement confirming that we were going to re-enter the drift and at that point I confirmed that the commitment was just to re-enter the drift. I have had to go back twice to Cabinet to get top-up funding to make sure we complete that and fulfil the commitment that we made - that is the limit of the funding.
"I know people will be saying 'just do a feasibility study, it won't cost much, or cost much more to go further'. Actually, we know from this whole project that what we thought might be time taken to do it and the cost taken to do it isn't actually the case. It takes a lot more time, it takes a lot more expense.
"I'm sorry that having fulfilled the commitment that we made, as regrettable as it is, I have access to no further funding to continue on to do stuff that we haven't previously committed to."
Little said he had told the families previously he was not committing to going further and that funding was "not a blank cheque".
Bernie Monk lost his son in the disaster. He is not on the reference group and has rejected the minister's claim.
He told RNZ's Checkpoint programme that 23 of the 29 families support a feasibility study on going further inside the mine, while two want the mine to be re-sealed.
"We started last night. As soon as I finished the meeting and Carol Rose (who lost her son at Pike River) and one of our experts which is Richard Healey (an electrical engineer) and our family, we went home. We did a call to the UK and Tony Foster answered it who was on the meeting last night - one of our experts - he got in touch with our man (another expert).
"He'd already been in touch with him before I rung him. That's how quick we've started on this."
The study will be be done free of charge by independent mine experts and will look into the costs of recovering the mine's main ventilation fan, identifying risk factors, and present them to the government.
Some families believe the fan may hold vital clues to what caused the disaster.
Monk said last week's statement by the reference group should not have been released.
"So they made decisions on behalf of the families because they thought that they had the complete mandate to make these decisions without going to them and asking them to vote on it and they made these decisions without the vote."
"No regrets at all at releasing it to the media because what it was was actually putting our truth out there to say we understand that it's just too hard. It will cost too much money to get further.
"It's not as simple as what some family members think or want other family members to believe."
According to her, the lack of support for it was because not everyone replied to emails or attended meetings.
Osborne, whose husband died in the explosion, tuned into last night's meeting and disputed the committee's claim that 23 families supported plans for a feasibility study.
"I don't understand how they say they have the support of 23 families when I know only about 13 families turned up for the meeting."
Monk said Osborne was incorrect.
Meanwhile, Andrew Little said going beyond the mine's roof fall was a huge undertaking, and there are other ways to look into the mine workings.
"Police have agreed to drill six extra bore holes and to send down scanners to try to retrieve exactly the evidence which some families are saying could be achieved by physically getting there."
The Pike River families hope their own feasibility study will be complete in the coming weeks.
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"I'm sorry that having fulfilled the commitment that we made, as regrettable as it is, I have access to no further funding to continue on to do stuff that we haven't previously committed to." just print a few more billion, that'll fix it.