Arlene Foster last night appeared to be digging in her heels and refusing to step aside as the First Minister designate in a move that could risk the future of devolution.
Senior DUP sources told the Belfast Telegraph that Mrs Foster should reconsider her decision and not put her personal pride before what was best for "the party and the people of Northern Ireland".
The sources said they believed that Sinn Féin wouldn't back down from its position of not forming a government with the DUP if Mrs Foster was nominated as First Minister before the inquiry into the 'cash for ash' scandal issued at least an initial report.
One DUP figure proposed a compromise move for Mrs Foster whereby she would accept another ministry in the new Executive, with the agreement that she would step up and become First Minister after the inquiry.
"I think that is an entirely reasonable compromise which doesn't involve Arlene being humiliated," the source said.
"It wasn't a good election for us, we can't ignore that.
"But this is a possible solution that I think Sinn Fein would go for and it would enable Arlene to regain lost ground."
Another high-ranking DUP source said that around a third of MLAs felt "angry and let down" by their leader.
But he said that Mrs Foster had strong support from her deputy, Nigel Dodds, and from senior party advisers.
"Arlene is determined to brazen it out and that is putting the future of devolution in jeopardy," he said.
"She was once an asset to our party, but she is now a liability.
"It's all about self-preservation - 'me, me, me, me'. I believe that's a major miscalculation.
"Sinn Féin won't back down on this one. Given their election result they've no reason to do so."
The sources said that just because this was Sinn Féin's position, the DUP shouldn't automatically oppose it.
"If Sinn Féin says it's Monday, it doesn't mean it isn't Monday and we all insist it's some other day," he said.
"I believe that Arlene and the team around her are leading us into a terribly bad decision."
Writing in yesterday's Sunday Life, Mrs Foster said she was determined to remain as party chief and lead the DUP into talks with Sinn Fein.
"To the 225,413 people who placed their trust in the DUP, I say thank you... people voted DUP in numbers not seen since 1984. Over 22,000 more voted DUP than in our record breaking election last May," she said.
"When the talks begin on Monday, I will seek to work with other parties to create the circumstances where we can not just get the Executive up and running, but do so in a way in which it will endure."
Mrs Foster was yesterday backed by party colleague Simon Hamilton.
He told the BBC that she had the "full support" of her party and the election result was an "endorsement" of her leadership. DUP Assemblyman Edwin Poots said it was vital that the parties worked together to get the Stormont institutions up and running as soon as possible.
Speaking to the News Letter, he warned against a return to direct rule. "Ourselves and Sinn Fein have to get over whatever problems we have and ensure that we can respond to the real needs of the public, particularly in terms of health, education and the other issues that really matter for the people of Northern Ireland," he said.
East Belfast DUP MP Gavin Robinson raised the possibility that Mrs Foster might step aside as First Minister designate until the Renewable Heat Incentive inquiry had concluded.
Speaking to Stephen Nolan on BBC Five Live, he stressed that it would be for her alone to decide.
"As a party that wants to see devolved government in Northern Ireland succeed we are not going to present impediments to progress but we are not going to have another party determine who is going to lead our party," he added.
Meanwhile, Sinn Féin president Gerry Adams has indicated that he is "open" to that idea.
Speaking to RTE's News At One he said:
"We don't want her to step down as leader of the DUP. That's a matter for the DUP. I respect Arlene, I have a meeting with her and Michelle O'Neill later on this afternoon. The question is to do with the cloud that's hanging over – and imagine half a billion of taxpayers money going to waste without explanation – we are asking Arlene to stand for one side to facilitate that but that's a question for her."
"We are open to all those ways of resolving these difficulties. Arlene has said she is innocent of all of theses charges, so she will want as much as anybody else to get that cleared up.
"We have said Arlene is not acceptable as First Minister until all of this is cleared up," he said.
Mr Adams was speaking ahead of a meeting with Northern Ireland Secretary, James Brokenshire, after the dramatic election.
"We are looking for them to come forward with a plan. We don't want plámás, we don't want to be patronised," he said when asked what he was hoping to get out of the meeting.