President Shimon Peres is to spend the coming days trying to convince the leaders of Kadima and Likud to find a way of power-sharing in a national unity government.
Senior members of Kadima and Yisrael Beiteinu are already conducting quiet talks on a possible coalition of the three largest parties in the Knesset.
However, Likud is still trying to form a right-wing government as a first step before offering cabinet places to Kadima.
Mr Peres has been holding consultations with representatives of all the parties in the Knesset. On Wednesday he said that “the problem is not only who will be the prime minister but what will be the policy of the state of Israel”.
His words were seen as an appeal to Kadima leader Tzipi Livni and Likud leader Binyamin Netanyahu to find a way to resolve their differences and cooperate in a unity government.
Likud are still insisting that, since the right-wing bloc has a majority, Mr Netanyahu should be prime minister. Kadima have slightly retreated from their demand that Ms Livni, as leader of the largest party, must be the premier.
“I am almost certain that we are going to have a national unity government,” said Kadima MK Othniel Schneller. “Tzipi should head it as the leader of the largest party but since the election results were inconclusive, a rotating premiership between her and Netanyahu is appropriate.”
Mr Netanyahu has ruled out a rotating premiership so far. However, he will keep a number of ministries vacant for Kadima if he forms a government.
Mr Netanyahu has still not secured a majority of Knesset members prepared to recommend him as prime minister. The small right-wing and religious parties have pledged support but he still needs Avigdor Lieberman’s Yisrael Beiteinu to achieve an overall majority.
Mr Lieberman has yet to indicate his intentions. Instead, he went on holiday to Belarus, leaving representatives to conduct talks with Kadima and Likud. Both parties have acceded to most of Yisrael Beiteinu’s policy demands.
“There are no great secrets here,” said Yisrael Beiteinu MK Danny Ayalon. “Both parties have offered us similar things, and what we will convey to the president is our desire for a stable government which will emphasise our civil agenda and electoral reform. It is not about the prime minister but about the government’s policies.”
Mr Ayalon noted that Likud had erred by promising Shas that it would be a central coalition member. Shas is now vetoing Yisrael Beiteinu’s demands for civil marriage and conversion reform.
While Likud is still vying for Lieberman’s support, no party apart from Kadima will recommend Ms Livni.
The Arab parties announced that, in the wake of the Gaza operation, they would support neither candidate. Meretz and Labour will not support Ms Livni, in protest over her acceptance of Mr Lieberman as a coalition partner.
Labour is in disarray following its worst-ever election result of 13 Knesset seats. On Saturday, former party leader Amir Peretz announced that he would be running for the leadership in an attempt to depose Mr Barak, and other senior members have already called for his resignation.