The Ulster Unionist leader, Mr David Trimble, has accused the DUP of engaging in political stunts and has said he is confident that the Assembly will elect First and Deputy First ministers.
The election is due to take place this morning.
However, the DUP has accused Mr Trimble of being anti-democratic and running scared of the electorate. The party said it was confident that, following a court decision yesterday, fresh Assembly elections would be held soon.
Mr Trimble was speaking after a DUP legal attempt to block the appointment of a First Minister failed in Belfast High Court. A lawyer for the Northern Secretary, Dr John Reid, accepted that he was under an obligation to propose a date for fresh Assembly elections.
The UUP leader said the DUP had lost the court action and said he believed the Assembly would soon proceed with the First Minister vote.
He accused the DUP of "playing games, pulling stunts, engaging in personal remarks rather than trying to improve the situation in this society".
DUP members engaged in a "series of cheap, snide comments" in the Assembly and were doing "anything they can think of, turning to every trick in the book in order to frustrate what is the clear wish of the greater number of Assembly members and also the greater number of people of Northern Ireland".
However, the DUP leader, the Rev Ian Paisley, welcomed the outcome of the court proceedings and said his party would vigorously pursue the Northern Secretary in the courts to make sure he would call an election.
"John Reid sent a man into court who had nothing else that he could do but say the law says he must call an election and he will call an election. If that is not a climbdown, I don't know what is.
"We are quite happy. The Secretary of State may surrender any time he wants, even if he wants to claim that as a victory."
Incoming SDLP leader Mr Mark Durkan accused the DUP of hypocrisy. "Remember among the people trying hardest to prevent the election of First and Deputy First Ministers are people who couldn't contain themselves when they were taking ministerial office just a few days ago."
Sinn FΘin MP Mr Martin McGuinness said it could be a good week for the peace process because the pro-agreement parties in the Assembly had "crossed the Rubicon together".
Speaking following the Alliance party's attempt to redesignate three members as unionist in order to re-elect Mr Trimble and save the peace process, Mr McGuinness said he was confident First and Deputy First Ministers would be elected within the next few days.
"This is a real opportunity to move forward and to give effect to the wishes of all the people of Ireland." He claimed that anti-agreement unionists had been outflanked.
"The DUP are very determined to bring down the Good Friday agreement and have, on this occasion, been out-manoeuvred by the pro-agreement parties. But we have to continue doing that."
He agreed that recent events had done little to enhance the international reputation of Northern politics but he added: "Anybody watching closely will see that all the cows were in one corner and they were all being led by Ian Paisley."
Alliance leader Mr David Ford said he was confident that First and Deputy First Ministers would be elected this week. He defended the decision of three of his party's Assembly members to redesignate.
Ulster Unionist Assembly member Mr Roy Beggs has said that re-electing First and Deputy First Ministers will increase prospects for stability in the North. He said he had not yet heard a "coherent argument" for voting against their re-election.
"Those unionists who oppose the re-election of the First Minister and Deputy First Minister following the commencement of actual decommissioning by the republican movement risk the unionist community being ostracised in the rest of the UK and in the wider world.
"As someone who wishes to ensure there is an ongoing process of decommissioning and a stable form of devolution, I do not understand how bringing down the Assembly at this time will help."