Monk said Little had reiterated his party's commitments, which are to get an independent assessment of the mine and re-enter it if it was declared safe.
Dunne also wanted a new assessment of all of the available information to check whether re-entry was feasible.
"With all the questions that have been asked, and the images that have been shown in public, he's had a huge change of support and it backing what the families are trying to do," Monk said.
Dunne was initially satisfied with a Royal Commission of Inquiry's findings into the 2010 disaster but has more recently become concerned about the ongoing leaks of new footage from inside the mine.
While he does not want lives to be risked to recover the bodies, he said that he thinks all of the available evidence should be thoroughly assessed to check that they do not contradict the Royal Commission's report.
"We need full disclosure," he said. "And someone must have all of this information."
The families will meet with the Greens and NZ First tomorrow. NZ First leader Winston Peters has previously said he will personally enter the mine to help recover the bodies.
The Government committed in February to exploring unmanned re-entry of the mine, possibly with the use of a drone. It has funded the mine's owner Solid Energy to carry out the task.
But the families are worried about the progress of this plan, especially as Solid Energy is scheduled to go into liquidation in April and hand over the mine to the Department of Conservation.
Prime Minister Bill English said he did not plan to meet with the families this week. But he stood by the commitments he made in February around unmanned re-entry.
"We will follow through on those undertakings. There will be some complicated conversations about ownership and what rights or obligations they have, but we will deal with those in the context of fulfilling the undertakings made to the families."
The families also filed their papers in the Supreme Court today for a hearing to appeal the decision to drop charges against former mine boss Peter Whittall.
Whittall originally faced 12 charges from health and safety watchdog Worksafe, but the charges were dropped and a deal was cut for Whittall to pay $3.41 million to the victims' families and two survivors.
The Supreme Court last month granted the families leave to appeal the decision.