The North's First Minister, Mr David Trimble, has called on the British Prime Minister to ensure the Provisional IRA implements a "clear and unambiguous" ceasefire.
He said Mr Blair had a responsibility to prove he was the guardian of the peace process. The British Prime Minister is expected to make a statement on the peace process and the paramilitary ceasefires before the House of Commons rises for the summer recess on Wednesday.
Speaking on the BBC's Breakfast With Frost, Mr Trimble questioned the sincerity of the Provisional IRA's apology for civilian deaths issued last week. "It certainly doesn't reflect the actions of the republican movement over the last couple of months, when they have been involved in serious violence in Belfast.
"From the point at which the Prime Minister became focused on the issue, the mainstream republicans have been winding down the violence. Apologies are fine, but it would be important that the apology is based on a change of approach and a change of heart by persons who are apologising.
"We have made it clear to the government that we expect the government to respond to this and indeed the Prime Minister has promised to do so." Mr Trimble said he hoped Mr Blair would this week clarify what his government planned to do.
"He is supposed to be the guardian of the peace process. He is the person who carries the legal responsibility for maintaining order here in Northern Ireland and I am calling on him to exercise that." Mr Trimble said people wanted the peace they were promised with the Belfast Agreement when it was signed over four years ago. "We want the paramilitaries to carry out the undertakings they gave to have a complete and unequivocal ceasefire.
"And, of course, to carry that through with the decommissioning of weapons and the eventual disbandment of the organisations. We didn't expect it would happen overnight but we want to see clear and unambiguous progress towards that. That is the responsibility of the paramilitaries and it is the responsibility of the Prime Minister to put pressure on them to do that. Let us all look forward to Wednesday in anticipation," said Mr Trimble.
Meanwhile, the North's Deputy First Minister, Mr Mark Durkan, warned Mr Blair not to damage the political institutions when he makes his statement.
The achievements of the Assembly were a true reflection of the success of the peace process, he said.
"Paralysing government punishes the public, not the paramilitaries. We must affirm the absolute primacy of the institutions. That means making clear that no party can respond to its own internal pressures or paramilitary wrongdoing by threatening to turn the lights out on our new democratic institutions of government," said Mr Durkan.
He said removing sectarianism was the greatest challenge facing the North. "It requires the establishment of trust where little exists. That can only be done by continuing to work together, to work the Agreement," he added.