The Ulster Unionist leader, David Trimble, claimed Mr Paisley had no intention of becoming first minister, despite being leader of the largest party.
Speaking on BBC Two's Newsnight, he said: "Ian's desire has always been to be in oppostion. He's always wanted to beat us too, but that's presented him with a problem.
"He can't remain in perpetual opposition as the largest party. That's a very difficult thing to do and I don't think he'll be able to sustain it very much longer."
Also on Monday, anti-Agreement Ulster Unionist Jeffrey Donaldson confirmed that he was threatened with a special party council meeting calling for his resignation.
Mr Donaldson, who again called for Mr Trimble to step down, said he told a meeting of the Ulster Unionists' Assembly group that he believed "that if the party was to rebuild for the next election, the message from the voters would have to be heeded".
Mr Trimble insisted that he would not stand down and said that Mr Donaldson's comments were not an accurate reflection of the "amicable" two-hour meeting.
"It is rather unfortunate that someone is producing a conclusion before we've had the discussion," he said.
Anti-Agreement Ulster Unionist David Burnside said that it had not been a good election but that the party had to try and rebuild.
"There is a strong feeling and a knowledge within the party all round that the UUP had better get united on policy, there is a split down the middle" he said.
Alliance Party leader David Ford told journalists at Stormont on Monday that reports of the demise of his party were "seriously exaggerated".
"We are back here to serve the people who elected us, to serve all those who wished to see a non sectarian future for Northern Ireland," he said.
Earlier on Monday, DUP deputy leader Peter Robinson told BBC Northern Ireland's Good Morning Ulster that his party was not "running away" from helping the government to find solutions.
"In fairness to the government... they have stripped back the Belfast Agreement to what they describe as its fundamental principles and what we will want to explore with the secretary of state is just exactly what he sees those fundamental principles as being," Mr Robinson said.
"We are not running away from helping the government to find solutions. They will find we have ideas and that we are positive."
Sinn Fein President Gerry Adams has written to Mr Paisley, requesting talks with the DUP.
The letter was hand-delivered to the party's Stormont offices.