Kadima came first in last week's elections, winning 29 seats of the 120-member parliament, and President Moshe Katsav has asked that Ehud Olmert, Kadima's chief and Israel's acting premier, be made prime minister.
But the centre-left Labour party, which came second with 20 seats, recommended to Katsav that its leader Amir Peretz be charged with forming the next cabinet.
Eitan Cabel, Labour's secretary-general, said Peretz had put "social issues at the top of the agenda" and thus should be premier.
Olmert's campaign focused on his plan to set Israel's final borders with or without Palestinian agreement by 2010, while Peretz's focused on raising minimum wages and increasing spending on welfare, necessitating Labour's control of the finance ministry.
Business community's fears
But Kadima has ruled out conceding either the premiership or the finance ministry post to Peretz, whose leadership of a costly trade union strike two years ago has made Israel's business community wary of his influence over the budget.
Peretz has been refusing to meet Olmert over the issue, Israeli media says.
After a 4% drop in the Tel Aviv stock market following the announcement of the election results, Olmert sought to reassure investors that there would be no change in Israel's broad economic policy, but conceded that the government "will have to change priorities regarding social issues".
"If [Olmert] wishes to set permanent borders, he'll have to set up a coalition with us"
Meretz party chief
The Labour party could conceivably put together a shaky coalition with right-wing parties, but is thought mainly to be holding out for a better deal in an Olmert cabinet.
Kadima, on the other hand, must look to the left-wing parties for support for Olmert's plan for setting Israel's borders by 2010, which involves Israeli withdrawals from Palestinian territories.
Yossi Beilin, leader of the left-wing Meretz party, which is also a potential coalition partner, said "a government with the right cannot push Olmert's plan forward. If he wishes to set permanent borders, he'll have to set up a coalition with us".
Other parties, which will meet Katsav on Monday or Tuesday, are reported to be still undecided on whom to back for the top job.
A coalition with Labour, Meretz and the Pensioners' List would give Olmert a narrow majority of 61 seats.
Kadima, however, believes at least four parties - the Orthodox Shas and United Torah Judaism factions, the ultranationalist Yisrael Beiteinu and the Pensioners - will recommend Olmert for the premiership.
Support from these parties would give Kadima a majority of 65 MPs and leave Labour out in the cold.
But the centre-left party has said that Yisrael Beiteinu can never be a reliable partner for Olmert's border plans.
Katsav has until April 12 to choose the next prime minister.
The new premier then has 28 days to form a government, with the possibility of a 14-day extension, before the new lineup is put to a parliamentary vote.