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Sinn Féin has called on Sir Jeffrey Donaldson not to "waste momentum" ahead of more talks to restore power-sharing.
The main political parties will meet the secretary of state again on Monday to discuss a potential financial package for Stormont.
North Belfast MP John Finucane said the Democratic Unionist Party (DUP) leader should "not waste the opportunity that is present".
On the table at the talks is a £2.5bn financial package for Stormont, which is based on the restoration of an executive.
It would include a lump sum to settle public sector pay claims and a new "needs-based" funding formula for public services, but party leaders have said it is not enough money.
The DUP quit power-sharing 22 months ago over post-Brexit trade arrangements.
In an email to members on Saturday, seen by BBC News NI, Sir Jeffrey said more work was needed and he would not be "sidetracked by calendar-led initiatives".
"I have no doubt the government will want to advance on their offer on Monday," he wrote.
"They will ultimately need to move on a substantive change to our funding formula."
'The time is now'
Speaking on BBC News NI's Sunday Politics, Sinn Féin's Mr Finucane said Sir Jeffrey needed to show leadership.
"What's clear this week is that there has been momentum," he said.
"It's clear that the time is now for Jeffrey Donaldson not to waste that momentum.
"If he prevaricates and allows this to drift then that will be to the detriment of everybody in this society."
Although there is no formal deadline in the talks, the UK government has been working to secure agreement before parliament breaks for Christmas on Tuesday, with a view to re-establishing the assembly quickly.
The parties are due at Hillsborough Castle on Monday to hold talks on the proposed financial offer.
Also, speaking on Sunday Politics, Eóin Tennyson of the Alliance Party said that what the DUP want to do and what they will do "are often two completely different things".
He added: "There's absolutely no reason why we can't be in the assembly pre-Christmas.
"I don't think the public will wear another Christmas where they are witnessing this kind of pantomime politics where they are traipsing over to Hillsborough for talks."
Any possible restoration of Stormont also depends on key discussions about the Windsor Framework and how Northern Ireland trades with Great Britain.
On Wednesday, Prime Minister Rishi Sunak said the government was ready to legislate to protect Northern Ireland's position and would work "at pace".
On Friday the DUP's 12 most senior members, known as party officers, met to discuss the situation, fuelling speculation an agreement was close.
In his email, the DUP leader said he had been able to identify "legislative measures and potential actions that would be in keeping with our negotiating objectives".
But he said they would be "measured against the tests in our manifesto, and that work for Northern Ireland and our place within the United Kingdom".
We do know there were lots of conversations on Saturday between all the key players about this process.
The UK government was expecting a phone call from Sir Jeffrey Donaldson to signal his party's intent to go back into the executive.
That didn't happen, the email to party members on Saturday made it very clear the DUP feels it's not there yet.
The focus will switch to Hillsborough Castle where those financial talks are due to resume again, with pressure growing on the secretary of state to make a call on all of this.
It's challenging for the government, they've built up a lot of momentum behind this process.
But there is a new date lurking on the horizon of 18 January, that is the date by which the secretary of state has to decide on another assembly election if power-sharing is not restored.
Where do the other parties stand?
Ulster Unionist leader Doug Beattie said "as it stands right now" his party intended to re-enter the executive if a new one is formed, rather than go into opposition.
"Yesterday we had our party executive meeting and we talked about this and we are intent and are moving towards going back into the executive," he said.
"I have to see what the outworkings are of the financial package, I have to see what the outworkings are of the programme of government, I have to see the outworkings of what a new reformed executive would be like and then I will make a decision on that."
Following poor election results in May 2022 the Social Democratic and Labour Party (SDLP) is not entitled to a place in the executive and will instead enter a formal opposition.
The party's Matthew O'Toole said that "unlike other parties" the SDLP "accepted the judgement of the electorate".
"If we are there by ourselves we will do the job and we look forward to doing the job," he said.
"If other parties wish to join us I look forward to working constructively and actively in partnership with them."
Jim Allister said if the DUP returned to Stormont, they would be "betraying" their seven tests on future post-Brexit arrangements.
But the TUV leader said he would re-enter opposition if the assembly was recalled.
He added: "I'm not in the executive, nor do I ever wish to be. I am an assembly member and if the assembly is there, I will be there as a voice of opposition."