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 26 June 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.
He said: "We will have no demands, other than that people are faithful to commitments and agreements that have been made.
"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 Fein 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 Fein shopping list."
The chaotic downfall of the DUP triumvirate
Speaking on Sunday Politics Northern Ireland on BBC One, he said: "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 and uncapable to deliver on people's rights."
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.
The DUP is craving stability after a turbulent two months that has seen former leader Arlene Foster resign after an internal heave against her and Mr Poots follow suit after he was also fatally weakened by a party revolt.
His resignation on Thursday night came after just three weeks in the post.
Mr Donaldson narrowly lost to Mr Poots in last month's DUP leadership contest to replace Mrs Foster.
