Update 5.26pm: Political leaders need to show courage to seal a deal restoring powersharing in Northern Ireland, the SDLP leader Colum Eastwood has urged.
He said he was encouraged that some progress had been made.
"Political leaders now need to be challenged to show some courage and get this across the line,” he said.
"There is too much at stake to allow our institutions to fall - the deadline may have passed but we still need to get the deal done.
"We cannot allow Northern politics to fall into a permanent status quo of huge frustration."
Cross-community Alliance Party leader Naomi Long said Northern Ireland Secretary James Brokenshire had told the parties he would "reflect" over the weekend on how to proceed, and allow further talks to go ahead in the interim.
She said she was "extremely frustrated and disappointed" that the latest deadline would not be met.
"This Assembly has often been criticised and held in low esteem," she said. "We do ourselves no favours when we continue to let deadline after deadline slip."
She said it was inexcusable that a row over the "name" of an Irish Language Act was halting progress.
"That is ludicrous after six months of negotiation and debate on the issue," she said.
Ulster Unionist leader Robin Swann expressed disappointment.
"It's not what we wanted and it's not what the people of Northern Ireland wanted. I believe that the majority of people voted at the last Assembly election for local politicians to take local decisions.
"Everyone should recognise that if a functioning Executive and Assembly are not in place and we go to direct rule, then they should not expect either to be back in place any time soon.
"It is the height of folly to allow any of the participants to hold a veto over the formation of the devolved institutions."
Nichola Mallon, SDLP North Belfast Assembly member, said people were rightly angry and frustrated.
"We always felt a deal could have been done by the deadline of today.
"The institutions were brought down in January, we have had two elections and negotiations since March and no new issue has emerged.
"But we are where we are, we will continue to play a constructive role."
Update 4.14pm: Northern Ireland's political leaders have missed today’s 4pm deadline to restore powersharing at Stormont.
Despite failing to reach agreement by what is now the fourth since the institutions collapsed earlier this year, negotiations look set to continue over the coming days to get a deal over the line.
It is understood Northern Ireland Secretary James Brokenshire has told the five main Stormont parties he will reflect on the situation over the weekend, allowing further talks in the interim, before making a statement at Westminster on Monday outlining his intentions.
Options open to him include setting a fifth deadline for the talks process, calling another snap Assembly election or reimposing some form of direct rule from London.
Update 2.30pm: The Democratic Unionists has said there will not be an agreement to form a new powersharing government in the North by today's deadline at 4pm.
The comments come despite the Irish and British governments saying "significant progress" had been made.
If a deal isn't reached by 4pm today, direct rule from Westminster could be imposed.
The DUP's Edwin Poots said: "The talks are continuing (but) obviously there's not going to be a breakthrough which will lead to nominations taking place today. The DUP were willing to nominate today and proceed, but Sinn Féin do not wish to nominate ministers."
Sinn Féin has warned the Democratic Unionists it is "make your mind up time" as the clock ticks down to a deadline to save powersharing in Northern Ireland.
Party negotiator Conor Murphy said "limited progress" had been made to bridge the gaps between Sinn Féin and the main unionist party and added that the DUP had to give much more ground if a deal to restore devolution was to happen.
The UK and Irish governments have claimed an agreement is still "possible and achievable", but as the 4pm deadline for the talks looms, Foreign Affairs Minister Simon Coveney and the North's Secretary of State James Brokenshire gave no firm indication that they would give the parties more time.
If the deadline passes and no extension is granted, Northern Ireland faces the prospect of some version of direct rule from Westminster or yet another snap Assembly election.
The key sticking point in the way of a deal has emerged as Sinn Féin's demand for an Irish Language Act, which would bestow official protection for Irish speakers.
The DUP is willing to legislate on the language issue, but only if Ulster Scots speakers are included in any Act, a condition Sinn Féin has rejected.
Mr Murphy said: "In our view it's now make up your mind time for the DUP. We have always said this can be done in a matter of hours - the issues are very clear, there are still gaps in terms of trying to establish a rights-based approach to these institutions working, as per the Good Friday Agreement.
"We want to close those gaps. There has been some limited progress in closing those gaps. We want to get this done, we want to get it done quickly and on a sustainable basis."
Mr Brokenshire said a number of issues remain outstanding.
"I believe a resolution can be found," he said. "And I'm urging the parties to continue focusing all of their efforts on achieving this. The UK Government will work with the parties toward their critical objective of forming an executive.
"But I've made clear to party leaders that it is for them to reach agreement...That prize remains achievable and remains my focus."
A scheduled meeting of the Stormont Assembly was earlier postponed amid intensive last-ditch negotiations to restore powersharing.
The Assembly had been due to convene at noon to begin nominating devolved ministers to a coalition government.
After a meeting of party whips, the start time was scrapped. It is understood the majority of the parties wanted to postpone the session until 2pm, but the DUP argued that the meeting should not proceed at all.
Mr Coveney said the last three days of talks were intensive and he was encouraged by the discussions.
"There still remain gaps to be bridged on some key issues," he said. "Like the Secretary of State I believe that an agreement is still within reach, an agreement that would allow a powersharing executive to be formed on a sustainable basis."
Mr Coveney said all parties are committed to the successful operations of all the political institutions in Belfast and everyone was convinced devolution was the best way to deliver for the people of Northern Ireland.
"With courage and goodwill this can be achieved and everybody here is continuing to focus on a positive outcome," he said.
The institutions imploded in January when DUP leader Arlene Foster was forced from office after Sinn Féin's then deputy first minister, the late Martin McGuinness, quit in protest at the DUP's handling of the Renewable Heat Incentive (RHI) - a scheme that left the administration facing a £490m overspend.
His move triggered a snap Assembly election in March.
A number of attempts to restore powersharing between the five main parties following that poll floundered, with three UK Government deadlines for a deal having already been missed.
British Prime Minister Theresa May's official spokesman said: "If, despite our collective efforts, it proves impossible to re-establish the executive, we will need to consider the options to ensure Northern Ireland has the political stability that it needs.
"In terms of what those options are, I wouldn't get into that at this stage."