The issue at the heart of the dispute is the DUP's refusal to put a timeframe on the devolution of policing and justice affairs until certain preconditions have been met.
One of these conditions is to allow Orange Order parades to march through Catholic neighborhoods.
Sinn Fein wants to keep the current system of having the Independent Parades Commission make a ruling on each parade route, and has stated that if an agreement cannot be reached on this issue that it will leave the Assembly.
Talks continued late into Monday night and were adjourned at 3.30 a.m. local time.
Irish Foreign Affairs Minister Micheal Martin told RTE that the situation was "serious," but that all parties involved were working hard to find a solution.