Sinn Fein has again called for Arlene Foster to step aside as First Minister and said that an independent investigation into 'cash for ash' - with the power to compel witnesses - is the only way forward.
Health Minister Michelle O'Neill said that a process was needed "which gets to the facts quickly, effectively, and with minimum cost to restore confidence in the political process".
But Sinn Fein has backed down on its previous demand that a public inquiry, rather than one behind closed doors, was needed.
Ms O'Neill said: "The quickest and most cost-effective way is a thorough, independent investigation which is robust, transparent, time-framed and led by a senior judicial figure from outside the jurisdiction.
"If Arlene Foster has any sense of the outrage and anger in the public then she will step aside."
Sinn Fein will bring a motion to the Assembly on the issue on January 16.
However, East Londonderry MP Gregory Campbell retorted that Mrs Foster was going nowhere and was under no pressure from within her party or the unionist community.
"The more Sinn Fein press for her to go, the more unionists will say: 'We need a strong woman and a strong First Minister'," he said.
Sinn Fein's national chairman Declan Kearney earlier said that if Mrs Foster didn't step aside for an "independent investigation" into the Renewable Heat Incentive (RHI) scheme, her position as First Minister would be "untenable".
Speaking on BBC Radio Ulster, he accused the DUP of "unvarnished arrogance" and said an Assembly election was possible.
TUV leader Jim Allister accused Sinn Fein of "bluster" and said it was significant that the party had capitulated on its previous demand for a public inquiry.
"Anything less provides a soft landing for its DUP partner which, I suspect, is Sinn Fein's real intent. No matter how it is dressed up, a mere "investigation" will be a vehicle for cover-up and whitewash," he said.
"Only a public inquiry, under the Inquiries Act 2005, would have the essential powers to compel witnesses and the production of documents."
Mr Allister said that a behind-closed-doors investigation would be "a toothless tool". He claimed that Sinn Fein was likely negotiating concessions from the DUP "as the price of blocking a proper public inquiry".
Ulster Unionist leader Mike Nesbitt was also sceptical over whether Sinn Fein would see through its tough verbal stance on 'cash for ash', or would back down after secret "horse trading" with the DUP.
Mr Nesbitt said that RHI was just "the latest in a long line of scandals, failures, and crises" that had plagued the DUP-Sinn Fein administration.
"They promised us a Fresh Start, but the only thing that is fresh about the current debacle is that it is not called Red Sky, or Nama, or the Social Investment Fund," he said.
Mr Nesbitt called on Mrs Foster to show leadership by accepting the principle of ministerial responsibility and resigning in order to restore public confidence in the political institutions.
"We need a judge-led, time-bound public inquiry under the 2005 Inquiries Act. It is high time personal careers played second fiddle to saving Stormont," he added.
Meanwhile, the British Government has rejected a call that it instigate a public inquiry into the RHI scandal, which is set to cost tax-payers almost £500 million.
Alliance leader Naomi Long had written to Secretary of State James Brokenshire and the Chief Secretary to the Treasury David Gauke demanding London's intervention.
But a Government spokesman last night said: "The operation of the RHI scheme is a matter for the Northern Ireland Executive. So it is right for the Executive and the Assembly to decide the form of any investigation or inquiry."
Mrs Long maintained that a judge-led public inquiry is the only way forward and dismissed Sinn Fein claims that it could take years as "nonsense".
She said: "This crisis can only be addressed by complete openness and transparency. Anything short of this would raise a number of questions and give the perception of something being hidden."
SDLP MLA Patsy McGlone also said that a public inquiry was the only viable choice.
"I want to see a situation where something is not conducted behind closed doors by a panel of hand-picked people and hidden away from public view," he added.