Sinn Fein's Martin McGuinness said he was "frustrated" with the DUP because of the lack of progress on devolving policing.
DUP MP Sammy Wilson said Sinn Fein had "thrown a hissy fit" and should "pull back from the brink" as it was possible to find a solution to the impasse.
"People said we could never deliver this assembly up and running again," he said.
"When we believed we had got the right things in place we went out and sold it - we have persevered with it and we have had the assembly running now for three years.
"So if we can get something we believe is workable that's the only criterion we will make - it's not about 'have we won or have we lost?' - if we get something workable we will sell it, and we will sell it hard because we want to see this place working."
The DUP and Sinn Fein - Northern Ireland's two biggest political parties - have been arguing for months over the transfer of policing and justice powers from Westminster to Stormont.
Sinn Fein want the completion of devolution to happen as soon as possible, but the DUP argue there must be unionist "community confidence" before powers are put in the hands of local politicians.
The "confidence" issue causing most division is over the handling of parades.
The DUP wants to scrap the Parades Commission, which puts conditions on some of the most contentious marches, but Sinn Fein has accused them of giving the Orange Order a talks veto.
Brian Cowen and Gordon Brown have travelled to Hillsborough Castle
Prime Minister Gordon Brown has postponed Tuesday's cabinet meeting to allow him to stay in Northern Ireland for more talks.
Speaking at a news conference in Downing Street after meeting Mr Cowen, Mr Brown said: "We believe that the problems that exist in devolving policing and justice are all soluble problems.
"We believe that it is right for Northern Ireland to move forward in this way now and we believe that together we can assist the completion of these talks."
Irish prime minister Brian Cowen said he was confident progress would be made.
He said: "We are going to Belfast to see in what way we can assist.
"It is very important that we get a successful outcome to these discussions and we believe that our going there should help bring a conclusion to the devolution issues."
The leaders left Downing Street to travel to Belfast just after 1500 GMT.
Neither the DUP nor Sinn Fein has made any comment since the end of Monday morning's talks, but speaking before the meeting, Mr McGuinness said he planned to be "positive and constructive".
The talks between the two parties, which became acrimonious last week, were considered to be critical.
On Monday, Mr McGuinness said: "The only agreement that is viable is an agreement between the DUP and Sinn Fein."
He said that within three months of the St Andrews Agreement, his party "took a historic and monumental decision" to support the police in Northern Ireland.
The leaders cannot agree over the transfer of police powers to Stormont
"Three years on, we are waiting for the DUP to deliver and honour their commitments," he added.
Mr McGuinness said he was "at pains" to make the assembly work, it was "his life's work" and he felt passionately about it. It should, he said, deliver for the whole community.
Before heading to Hillsborough Castle, the DUP's acting first minister Arlene Foster said her party was ready to deal with the outstanding issues.
She said she hoped the outcome of the talks would mean "policing and justice powers that are sustainable, and will not involve the two prime ministers coming back to Northern Ireland, with the greatest of respect to them, for quite a long time".
The Ulster Unionists and the nationalist SDLP have both called for the talks to be widened out to include all the parties rather than just the DUP and Sinn Fein.
Ulster Unionist leader Sir Reg Empey said: "We will not be prepared to accept a piece of paper if it was negotiated by others in our absence."
SDLP assembly member Conall McDevitt said it was important for all the parties to be involved, "otherwise what we will have is old IRA men and people with the best interests of the Orange Order and maybe not of all the people of Northern Ireland, trying to come to conclusions which ultimately will only serve a very narrow section of the community".
Alliance Party leader David Ford, who is thought to be a possible future justice minister, said it was "unfortunate but necessary" that the prime ministers had to intervene "because politicians here aren't grown up enough to take their own decisions".
If Sinn Fein was to decide Mr McGuinness should resign, the joint nature of the roles of first and deputy first ministers would mean that Peter Robinson would also be forced out of office, collapsing the executive.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.