Sinn Féin has said it will have “no demands” for facilitating the appointment of a new first minister, other than a commitment to agreements already made.
The party's MP for South Down Chris Hazzard has said there is "no Sinn Féin shopping list" for fresh concessions from the DUP.
DUP party officers have set a date of June 26th to elect a new leader, with the newly appointed First Minister Paul Givan having been told he will be expected to step down as part of the transition.
Appointing a new first minister will require Sinn Féin to renominate Michelle O'Neill as deputy first minister in order to keep Stormont up and running.
Anger at a UK Government pledge to grant Sinn Féin a key concession on Irish language laws led to Edwin Poots’s resignation as DUP leader after just three weeks, but Mr Hazzard says his party will have no further demands.
“We will have no demands, other than that people are faithful to commitments and agreements that have been made,” he said.
“So 15 months ago we all restored devolution on the basis that the commitments made in St Andrews 15 years ago, that they will be finally realised in legislation.
“That’s where all the parties bar the DUP are at, it’s not just Sinn Féin saying this. That’s where I believe certainly the vast majority of people in the public are.” There’s no shopping list and there’s certainly no Sinn Fin shopping list.”
Mr Hazzard denied Sinn Féin were culpable for the latest Stormont crisis, by bypassing the DUP and calling on Westminster to legislate for the Irish language.
He said the DUP were “caught in a cul-de-sac of their own making”.
He told Sunday Politics Northern Ireland on BBC One: "When people are looking in at what's happening in the DUP, if it wasn't so serious it would be farcical.
“There’s no doubt that the DUP have caught themselves in a cul-de-sac all of their own making in recent years. I think they have been unwilling to deliver on people’s rights.
"For all the shouting about the protocol and Brexit, we have to remember it was gay conversion therapy that brought Arlene Foster down, and it was delivering on Irish language rights that brought Edwin Poots down. That's a really sorry state of affairs."
The current frontrunner for the DUP leadership is Lagan Valley MP Jeffrey Donaldson.
His future could be known as early as Tuesday, the deadline for nominations in the leadership contest, if he is selected unopposed.
Chastened by the bruising campaign that Mr Poots won, the party hierarchy’s preference is for an uncontested appointment this time round. However, it remains to be seen whether Mr Donaldson, if he does run, will face any challengers.
That will soon become clear, after DUP party officers set a deadline of noon this Tuesday for nominations in the leadership contest to be submitted.
A meeting will be held next week, on June 26th, at which the next DUP leader will be selected.
During the last campaign Mr Donaldson vowed to quit as an MP in Westminster to return to the Assembly and take up the role of first minister.
If that were to happen, it would mean a Westminster byelection in Lagan Valley, a seat he has held since 1997.
One possibility is that he could seek to replace Mrs Foster as the MLA for Fermanagh and South Tyrone when she steps down.
However, with the DUP currently in such turmoil there are doubts whether the party would want to voluntarily trigger a by-election in Lagan Valley any time soon.
Another option could see Mr Donaldson wait until just before the next Assembly election, scheduled for May 2022, before he resigns his parliamentary seat.
In that scenario he might appoint a temporary first minister to fill the role in the interim. -PA