Mr Trimble and Mr Durkan were not re-elected
Secretary of State Dr John Reid has been holding talks with the main political parties after the assembly failed to re-appoint David Trimble as First Minister.
Dr Reid said "rapid and substantial progress" has been made and the institutions could still be saved.
But the five-strong Alliance Party has so far resisted pressure to take action to avert the crisis.
The latest crisis was triggered after two of Mr Trimble's party refused to back him in a vote to return him to his position, following his resignation four months ago over the refusal of the IRA to decommission.
Under devolution legislation, that means the power-sharing executive can no longer function and faces collapse at midnight on Saturday.
Dr Reid must then decide whether to suspend the institutions - which under the rules would allow another six weeks to solve the problem - or call new assembly elections.
Dr Reid said talks would continue and he hoped to make an announcement on Saturday.
Alliance Party hopes
Just over 70% of assembly members in total backed Mr Trimble.
However, under assembly rules he needed majority backing from both the nationalist and the unionist voting "bloc" - and at just over 49%, the required unionist majority was lacking.
A possible way to save the assembly would be for the neutral Alliance Party to "redesignate" as unionists - meaning its votes could be counted in the unionist bloc and tip the balance of a new vote in Mr Trimble's favour.
But on Friday night party leader David Ford refused to say whether it would redesignate.
He demanded first radical reforms of the voting structures, to ensure future assembly votes were not carried out along sectarian lines.
He said: "What we are looking for is a reform of the system so that a majority of pro-Agreement votes counts."
Mr Trimble had hoped to return to his position following last week's decision by the IRA to begin disarming.
On Friday he said that although he was not seeking the redesignation of other parties to secure his re-election, he would accept such a gesture if it was offered.
Disciplinary hearing
If the vote had succeeded it would also have seen the new leader of the Nationalist SDLP, Mark Durkan, re-elected as deputy First Minister.
The Irish Taoiseach, Bertie Ahern, said he was disappointed at the failure to elect a first and deputy first minister.
He said his government was in "active and continuing contact with the British Government and the pro-Agreement parties".
A spokesman for the US State Department in Washington said they would also be discussing the situation with the British and Irish Governments.
The leader of the anti-Agreement Democratic Unionist Party, Ian Paisley, called for fresh assembly elections.
He said the result of the vote was a "bloody nose for all who took part in the charade of decommissioning".
Sinn Fein's Martin McGuinness said the result was a "great disappointment" to everyone, and also called for assembly elections.
Peter Weir and Pauline Armitage, the two UUP members who failed to support Mr Trimble, now face a disciplinary hearing within the party and could be expelled.