The United Iraqi Alliance, the Shia bloc that won a majority of seats in the new assembly, has said that it intends to hold talks with Kurdish and Sunni groups.
The possibility that minority Sunni factions, who won 44 seats in the 275-seat parliament, could participate in the government has been an encouraging sign for some observers.
Nonetheless, as the jockeying for key governmental positions begins, it takes place against the backdrop of further bombings, concern is growing over the fate of a kidnapped US journalist and reports that few Iraqis fully understand the constitution they voted to ratify in October.
The United Iraqi Alliance won 128 seats in the election and are, even with the 53 seats gained by Kurdish parties, three seats short of a majority needed to elect a president and push through reforms.
Many Sunnis allege that the election was rigged; but the Shia faction says it would form a coalition with Sunni groups if they actively discouraged further terror attacks.
Nizar al-Samaraei, an Iraqi analyst, speaking to Aljazeera in Baghdad on Saturday has said Iraq is in need of a national unity government that includes all sections of the society.
It is not clear who will replace
Talabani (L) as president
"A single group or even an alliance of two groups can never express all the expectations and ambitions of the Iraqi people," he said.
"At least four groups should form an alliance in order to meet Iraq's needs of safety and services."
The US, eager to scale down its military presence in Iraq, refused to comment directly on the debate among the Iraqi parties.
However, Washington is believed to be disappointed that the secular list headed by Ayad Allawi, the former prime minister, saw its number of seats fall from 40 down to 25.
The results of the election indicated that Iraqis voted along ethnic and religious lines rather than on ideological grounds.
Sean McCormack, the US state department spokesman, played down any religious division.
"Over time, you see in democracies that those sorts of things gradually start to fade and more and more you see groups come together around interests and issues," he said. "That's really a vestige of Saddam Hussein's era where he ruled by dividing and conquering."
According to reports, many Iraqis do not fully understand the national constitution ratified in a referendum on 15 October and there are fears that the document could exacerbate existing ethnic tensions.
The constitution divides the country into three largely self-governing regions along ethnic and religious lines.
As each region will control future oil discoveries, the Sunni minority, which lives in the oil-poor centre of Iraq, may not benefit equally from the riches, and some say this is the formula for civil war.
Karim Shamaa, an oil consultant, said: "The articles on oil are so blurred and so unclear and so contradictory, that it will never work."
Under a deal to win Sunni support for the constitution, the new parliament must consider amendments to the document in the first four months.
If the changes are approved by the legislature, then they will be voted on by the public in a new referendum.
Meanwhile, away from the political wrangling, violence continued. On Friday, two US marines were killed in a suicide attack in western Iraq and five members of President Talabani's staff were injured in a roadside bomb in the north of the country.
Fears are growing for the safety
of US hostage Jill Caroll
Fears were also growing for Jill Carroll, the kidnapped American journalist, after a deadline for her release passed on Friday.
Kidnappers had threatened to kill the 28-year-old reporter for the Christian Science Monitor unless all female inmates in Iraqi jails were released by Friday.
On Saturday, representatives from the US Muslim group, the Council on American-Islamic Relations, flew into Iraq in an effort to secure Carroll's release.