Roundtable sessions on Strands One and Two of the Stormont talks featuring all the participants have been suspended for two weeks to allow the parties to hold bilateral meetings, as well as discussions with the chairman, Senator George Mitchell, on the way forward.
Nationalist sources inside the talks said they expected to know by Christmas if the Ulster Unionists were "serious" about reaching an agreement.
Participants will discuss Strand Two, covering relationships between North and South, in bilateral meetings for the first three days of this week. Next week there will be three days of discussions on Strand One, covering internal arrangements in Northern Ireland. The following week, a three-day plenary will review progress.
Senator Mitchell is expected to meet all the parties separately over the next fortnight. He will be seeking to identify areas of contention and agreement with a view to presenting a paper to the plenary as a basis for further discussion.
Speaking on Sky TV yesterday, the SDLP leader, Mr John Hume, said Mr Mitchell would be "taking real charge of the negotiating situation. And we're hoping now that we will really start to get down to the real talks of those talks, to reach an agreement".
Mr Hume did not expect any walk-outs by the unionists. "I expect people to be applying themselves to reach an agreement."
On the issue of Bloody Sunday, he was asked if an apology would "make a difference" to nationalists. "The British government should take the necessary steps to establish the truth of what happened on that particular day," Mr Hume said.
The Sinn Fein president, Mr Gerry Adams, is expected to meet the British Prime Minister, Mr Tony Blair, at Downing Street in the near future.
Meanwhile, Mr Trimble has criticised the Government over Articles 2 and 3 of the Constitution, in an interview with the Belfast-based News Letter.
"The territorial claim must disappear and not simply be qualified by reference to consent. Such an offer is no improvement," he said.
"The Irish Government has got to come into the real world and respect the rights of the people of Northern Ireland," the UUP leader said.
The Northern Secretary, Dr Mo Mowlam, said at the weekend she had been told the current IRA ceasefire was stronger than the previous one.
Dr Mowlam, who briefed White House officials in Washington last week, told Radio Ulster: "What we've been doing is saying that when you have a talks process there is stress across the board - all parties find it difficult in negotiations.
"And from my readings, as I am sure the Americans in the administration are fully aware, it has not affected the nature of the IRA ceasefire. That, I am told, is stronger than the first (ceasefire), and it hasn't affected Sinn Fein's position in the talks."