The Northern Ireland secretary has said there is “no appetite for another election” after the deadline to form a new power-sharing executive in Northern Ireland passed.
James Brokenshire said there was a “short window of opportunity” to strike a deal.
He added that he will be making a full statement in Westminster on Tuesday.
Negotiations collapsed on Sunday after this month’s snap election, ahead of Monday’s 16:00 BST deadline.
The two biggest parties, the DUP and Sinn Féin, blamed each other.
Sinn Féin was “not in agreement-finding mode” during talks to save Northern Ireland’s political institutions, Democratic Unionist Party (DUP) leader Arlene Foster has said.
Sinn Féin’s Michelle O’Neill said the DUP did not have “the right attitude”.
On Sunday, her party said it would not be nominating a candidate for the position of speaker or for the executive office.
With no deal in sight, Monday’s Northern Ireland Assembly session to appoint a first and deputy first minister was suspended.
As there is no executive in place to agree a budget, responsibility for Stormont’s finances is set to switch to the permanent secretary at the Department of Finance on Wednesday.
Northern Ireland Secretary James Brokenshire now faces a choice between calling another election, imposing direct rule from Westminster or playing for time in the hope some compromise can be found.
He said progress had been made on a “number of issues” in the talks but that there are “significant gaps between the parties, particularly over issues surrounding culture and identity”.
“I believe there remains an overwhelming desire among the political parties and the public here for strong and stable devolved government.”
He added: “I don’t think there is any appetite for any further immediate snap election. The public have only recently gone to the polls and delivered what, I think, is a very clear message in seeing and wanting to see devolved government back on its feet again.”
Irish foreign minister Charlie Flanagan said he was “extremely disappointed” at the lack of a deal.
“It is particularly concerning that a vacuum in devolved government in Northern Ireland should now be occurring just as the island of Ireland faces up to the many serious challenges represented by the UK exit from the EU.”
Earlier, Mrs Foster said she doubted whether Sinn Féin had ever been serious about reaching agreement during three weeks of talks.
“These talks have failed because there wasn’t a recognition of everyone’s mandates,” said the former first minister.
“The government of Northern Ireland is not a game – it is very serious.
“The decision of Sinn Féin not to nominate today and block the creation of a new executive is very regrettable and damaging to all the people we represent.”
Analysis: Mark Devenport, BBC News NI political editor
It was “simply the worst” talks process ever, said the Ulster Unionists’ chief negotiator.
And there is not a lot of people rushing to contradict him.
Blame will be thrown in different directions.
DUP’s deputy leader Nigel Dodds added: “The jury us out on Sinn Féin’s position at the moment and I think they will have to be tested. We are certainly up for devolution.”
Sinn Féin’s Gerry Kelly denied claims the party were not interested in getting devolution back on track.
“No-one has worked harder than ourselves to try and get these institutions to work. But, they have to work on the basis of equality and on the basis of respect and that has not been there.”
Earlier, Mrs O’Neill, Sinn Féin’s leader at Stormont, said her party would be “standing firm”.
Sinn Féin wants to see the implementation of legislation to give official status to the Irish language, a bill of rights and a way forward on dealing with the legacy of Northern Ireland’s Troubles.
“Unfortunately, the DUP maintained their position in relation to blocking equality for citizens – that is a problem,” said Mrs O’Neill.
“We said consistently that we need to see implementation of issues that were previously agreed.
“That is a major stumbling block – the DUP didn’t approach it with the right attitude.”
Mrs O’Neill also said UK government representatives “did not play their role” in the way they should have done.
Reacting to Mr Brokenshire’s statement, SDLP leader Colum Eastwood said agreement was still possible and that gaps between the parties were closing.
“Over the course of the last few weeks we reached common ground on a number of key issues and closed gaps on others.
“Even on issues which seemed intractable, like Brexit, significant progress was possible.
“We must all seek to occupy and expand that common ground, not see it scorched.”
He called for a new talks process with an independent chairperson.
Ulster Unionist Party MP Tom Elliott said the talks process was “shambolic” and accused the secretary of state of letting them “drift along”.
“He needs to get a grip of the situation which now confronts us and should not allow the largest parties to dictate the agenda or timing of progress.
“We need to change the way we do business so that the devolved institutions remain stable.”
Alliance Party leader Naomi Long said parties now need to “get serious” over striking a deal.
“There needs to be drive shown by the parties involved – those holding things up cannot be allowed to drag things on any longer.
“While that has been frustrating and disappointing, there still exists an opportunity if people are committed to the talks.”
People Before Profit’s Gerry Carroll said MLAs’ salaries should be cut if Stormont is mothballed.
The political deadlock came after a snap election on 2 March brought an end to Stormont’s unionist majority and the DUP’s lead over Sinn Féin was cut from 10 seats to one.
Under Northern Ireland’s power-sharing agreement, the executive must be jointly run by unionists and nationalists, with the largest party putting forward a candidate for first minister.
The party said it would not share power with Mrs Foster as first minister until the conclusion of a public inquiry into the Renewable Heat Incentive (RHI) scheme.
Mr McGuinness, who had been suffering from a rare heart condition, died last week at the age of 66.
Stormont talks: ‘No appetite for fresh election’}