DUP and SDLP leaders said they are encouraged by two hours of "intensive" and "realistic" discussions between the two parties yesterday afternoon.
Both sides plan to meet the British Prime Minister, Mr Major, in the near future, and the DUP leader, the Rev Ian Paisley, was optimistic that some issues could be agreed by further dialogue and by "different ways" of doing things.
The meeting took place in the Dunadry Inn outside Belfast, and afterwards Dr Paisley said that the talks covered "the whole sphere of issues" and "this was a very valuable meeting for us all."
The SDLP leader, Mr John Hume, said they had a "very intensive discussion on the current overall situation".
"We had areas of agreement, we had areas of disagreement but we are going to continue the dialogue in order to achieve our ultimate purpose of lasting peace and stability for all our people."
Pressed about his party's opposition to elections, Mr Hume said that "our position has been made very clear on the current situation and Ian's position and his party's have been made very clear as well."
Dr Paisley commented: "We must make it clear, we didn't go in to congratulate each other. We went in to face big problems and big gulfs and big chasms, and we faced them.
"There was disagreement on some, there was some agreement, there was also a position where we thought that some [issues] could be agreed by more discussion and perhaps by other ways of doing it."
He said it was "a realistic meeting in a very realistic situation". He added that it was "a serious time and we thought that we should meet and face up to issues, which we did."
On his way into the talks the DUP deputy leader, Mr Peter Robinson, was asked if he could persuade the SDLP to agree to elections. "I think the SDLP should be persuaded towards an election," he said. "I don't think they've anything to fear from an elective process.
"We need to persuade them that we aren't intending to put anything off by way of this process. As far as we are concerned, as soon as the results are declared we are happy to sit down with them and start a negotiating process."
He added that every party should elect negotiating teams and those teams should choose for themselves "when and where and how they might get into negotiations. In those circumstances, very clearly we would not be sitting down with Sinn Fein, particularly a gun toting IRA."
Mr Hume was accompanied at the meeting by the SDLP deputy leader, Mr Seamus Mallon, Dr Joe Hendron, Mr Eddie McGrady and Ms Brid Rodgers.
Dr Paisley was accompanied by Mr Peter Robinson, the Rev William McCrea and Alderman Nigel Dodds.
