The Observer reports that unionists loyal to Britain have threatened to walk out of the talks on Sunday unless the Irish Government publicly drops its constitutional claim on Northern Ireland.
It says that lawyers working for the Ulster Unionist Party have held secret talks with Irish officials on changes to the republic's constitution.
The paper quotes an Irish source as saying that the meeting also tried to allay unionist fears about north-south bodies evolving into an embryonic all-Ireland government.
The Sunday Times criticises the Irish Prime Minister, Bertie Ahern, over what it calls "heavy pressure put on Tony Blair to demand further concessions from unionists."
It goes on to say that Mr Ahern is stuck between appeasing his own party, Fianna Fail, and getting on with the peace process.
However the paper adds: "Ahern has the capacity to manage his own party and public opinion and he proposes a non-coercive approach, which has the support of the vast majority of public opinion."
A more upbeat message comes from the Sunday Mirror which headlines its story: "I'm full of hope - Blair predicts Ulster peace deal by Thursday." The piece concentrates on the teamwork it says has been show by Mr Blair and Mr Ahern.
The paper goes on to report that an "urgent meeting" between Mr Ahern and the Ulster Democratic Party Leader, Gary McMichael, would happen in the next two days. It also says that the Northern Ireland Development Minister, Paul Murphy, is insisting that a peace settlement will be in place by Thursday.
"Hopes rise in final push" is the headline from the Mail on Sunday. It also takes the line that the process is "full of hope" as Thurday approaches. It adds that the talks at Stormont are "moving by the minute" despite the unionists' meeting call.
The paper also says sources are claiming that the government has taken part in a "bogus lowering of expectation ahead of a victory." The paper described this as "a standard Number 10 technique."
In a separate section the paper reports that IRA leaders are believed to be meeting on Sunday to approve a four-month extension to the current ceasefire.
The Observeris the only paper to carry Northern Ireland on its front page but as the week goes on and the deadline gets nearer, the focus in the UK press is sure to examine this issue even more closely.
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