The DUP and Sinn Fein had been sharing power for nearly 10 years when the devolved government collapsed over a botched renewable energy scheme costing millions of pounds.
In a speech, DUP leader Arlene Foster called for the immediate return of devolved government and time-limited legislation to deal with culture and language issues.
"I am proposing that we restore an Executive immediately", she said.
"We think we should be able to come to a determination pretty quickly whether Sinn Fein want to go back into government".
The British and Irish governments, who are facilitating the talks, have warned that failing to reach an agreement would have "profound and serious" implications and limit Northern Ireland's influence in Brexit negotiations, although no one is forecasting a return to serious violence.
For the first time in a decade, the region faces direct rule from Westminster.
"Certainly for our part we do".
Disagreement on issues such as same sex marriage and dealing with the past have been hampering progress with the biggest obstacle to a deal DUP opposition to the Sinn Fein demand for a standalone Irish language act.
"It is the first time that I have ever heard a leader of the DUP commit to legislating for the Irish language". Now we stand on the cusp of a new century for Northern Ireland.
"For over 10 years Martin McGuinness maintained the Executive while seeking to have the outstanding commitments of the agreements implemented with respect and equality for all", she said.
"The DUP's refusal to afford Irish language speakers in the North rights which speakers of Welsh, Gaelic and Irish enjoy across these islands is shameful and speaks more of our divided past than of the shared future Sinn Féin wishes to see".