Sinn Féin has said it will meet both the Irish and UK governments on Thursday to discuss the latest failed bid to restore power sharing.
The talks ended on Wednesday after the DUP declared there was "no current prospect" of a deal.
Both parties had been in negotiations in a bid to end the 13-month stalemate at Stormont.
Arlene Foster said the talks failed due to disagreements with Sinn Féin about legislation for the Irish language.
She said that Sinn Féin's "insistence on a stand-alone Irish Language Act means that we have reached an impasse".
The party's Stormont leader, Michelle O'Neill, accused the DUP of having "collapsed" the talks process.
Speaking to the BBC's Talkback programme, Shaun Woodward, former Labour NI secretary, said the situation "genuinely poses real risks for the future of peace in Northern Ireland".
He said: "Theresa May risked everything when she decided to do a deal with the DUP, it was a catastrophic error of judgement to try to play with the politics of Northern Ireland by dabbling in what is a very fragile ecosystem."
One possible outcome of the talks impasse is a return to direct rule in Northern Ireland.
However, Deputy Irish Prime Minister Simon Coveney told RTÉ on Thursday "the Irish government is not going to be allowing a situation where we move easily into direct rule".
Sinn Féin's Conor Murphy, told the BBC's Good Morning Ulster programme that the DUP's statement on Wednesday was "very depressing".
"We were working on the basis that accommodation had been reached," he said.
"We were expecting the DUP to go off and close the deal in relation to that, but they didn't do it.
"So, we want to talk to the two governments today, and I don't doubt we will be in touch with the other parties to see where this goes from here," he added.
The DUP MP Gregory Campbell said a "narrative" came to light over the weekend that there was a "possibility of a deal - despite us privately and publicly saying that if there was, it couldn't include an Irish Language Act".
"Now, where the future has to lie is in trying to pick up the pieces and see if agreement is doable," he said.
"We are going to have to get the budget passed next week, then we're going to have to clear the air, and get back as political parties at some point in the future.
"One thing I agree with Sinn Féin, they said - 'these issues aren't going away' - and they are right.
"But what they also need to understand, is neither are we."
Analysis: Enda McClafferty, BBC News NI political correspondent
Picking up the pieces after another failed talks process is not where both governments expected to be.
But, they will now have to plot the next move after negotiations between the DUP and Sinn Féin broke down on Wednesday.
The parties have clashed over who was to blame.
Sinn Féin will set out its response to the collapse after meeting both governments.
While the DUP have made it clear they want Westminster to take charge and legislate for a budget, it is now up to London and Dublin to decide how the talks process can be salvaged.
When asked about the prospect of direct rule from Westminster, SDLP leader Colum Eastwood said any potential future governance of Northern Ireland would require "strong" involvement of the Irish government.
"The rest of us cannot be left at the whim of Sammy Wilson, or Ian Paisley Junior or Gregory Campbell," he said.
"That just cannot be allowed to happen."
Meanwhile, the Alliance Party leader Naomi Long said it was a "very, very sad day for Northern Ireland, the institutions and the people of Northern Ireland".
She added: "It is a crying shame we are sitting here in the wake of yesterday's chaos, looking at a budget being imposed from Westminster, rather than engaging in the difficult job that is governing Northern Ireland ourselves."
Ulster Unionist Party leader Robin Swann has called for the publication of any progress made between the two large parties "so the people of Northern Ireland can see where the finishing point was".
Despite Wednesday's announcement from the DUP, the Northern Ireland secretary signalled that a deal remained possible.
"I believe the basis for an accommodation still exists," said Karen Bradley.
However, she is expected to come under pressure to legislate for budget at Westminster.
The Shadow Secretary of State Owen Smith told the BBC it was "enormously disappointing" that the talks collapsed and called for "some clarity" to be provided about "where progress was made, and where agreement was possible".
"I don't think it's good enough, frankly, to continue with this veil of secrecy with people unable to be certain about where there was agreement or where there wasn't," he added.