Reforming a contentious cross-community voting mechanism in the Northern Ireland Assembly represents a key negotiations sticking point, a DUP source said.
The region’s largest unionist party is also seeking changes to ensure the devolved parliament is more stable in future and cannot be readily pulled down.
On Thursday, Secretary of State Julian Smith said the party was thwarting a pre-Christmas return of powersharing institutions which have been on ice for almost three years.
DUP MP Sammy Wilson said: “We wanted to move forward and have as quick an outcome to these talks as possible but we will not be simply accepting any kind of deal just to satisfy his desire to have the Assembly up and running.”
He told the BBC’s Radio Ulster he did not want the process to drag on.
He added: “We are not prepared to go for a quick fix which then falls apart at some point in the future.”
A petition of concern designed to protect minority rights in the Stormont institutions has been misused to veto significant change like abortion, critics believe.
Discussions around change surround making it more difficult for one party to use the petition to block controversial legislation.
The parties are also considering amending Assembly rules to make it harder to collapse the institutions.
The late deputy first minister Martin McGuinness’ resignation in protest at the DUP’s handling of a green energy scheme ended powersharing in 2017; lengthy negotiations have failed to produce a resurrection.
Extra money for the region’s crisis-hit health system and other projects is on offer from the British Government if a deal can be struck.
The DUP has said it has not seen the details of any financial offer.
Sinn Fein president Mary-Lou McDonald said nobody was expected to sign up to a blank cheque but it was in everyone’s interests to make progress.
She said: “Now is the time of truth, we need to get the institutions of government back up and running.”
She added: “We want every party to step up to that ground with us and recognise that at a time of acute political challenges, at a time when clearly in particular our health services are under considerable strain, there is now a duty and obligation for people to lead.
“This is about leadership. This is about creating a deal that recognises people’s rights and allows us to get the institutions back up and running.”
She said: “There comes a moment when political parties and political leaders have to lead and that is the moment we are at now.”
East Antrim MP Mr Wilson said it was strange the UK Government had been unwilling to blame Sinn Fein for its role in the crisis but suddenly Mr Smith was “pointing the finger” at the DUP.
“If he had a rush of blood to the head yesterday hopefully now he is going to go back to being impartial on these issues,” he said.
Mr Smith met DUP representatives in London in a bid to secure a breakthrough in the powersharing talks.
He tweeted: “I have had a positive meeting with members of @duponline this morning at Westminster. I am confident that we can move things forward.”
The DUP has said it would not be forced into a deal which is not fair and balanced.
The Government has imposed a deadline of next month for agreement or fresh Assembly elections could be called.