The demands from the United Iraqi Alliance could spell trouble for pro-U.S. Prime Minister Ayad Allawi, a secular Shiite whose ticket is running second. Allawi had been seen as a possible compromise candidate if the Shiites and their allies don't win two-thirds of the 275 National Assembly seats.
"The Alliance would like to get either the position of the president or the prime minister and it prefers that it be that of the prime minister," Redha Taqi, a top official in one of the coalition factions, told The Associated Press.
Partial results have shown the Alliance, which has links to Iran and is endorsed by powerful cleric Grand Ayatollah Ali al-Sistani, with two-thirds of the 3.3 million votes counted so far in the race for the National Assembly.
The list headed by the U.S.-backed Allawi had about 579,700 votes, or 18 percent.
In other developments:
The election returns, released Friday, are only a partial count from 10 southern provinces in Iraq's Shiite heartland, where the Alliance had been expected to do well. But if it amasses a large vote count in the south, Allawi will have a tough time making up the difference in the other provinces.
Results from much of Baghdad, the Sunni Arab provinces of north-central Iraq and the Kurdish provinces in the north have yet to emerge.
The Iraqi election commission released no new election returns Saturday, but predicted it would announce final vote totals by Thursday.
Kurdish leader Jalal Talabani has announced his candidacy for the largely ceremonial presidency. The Kurds are expected to win a sizable bloc of assembly seats and could form an alliance with the Shiites, both of whom suffered under Saddam Hussein.
A group of Sunni Arab parties that refused to participate in the election said Saturday they want to participate in the drafting of a permanent constitution — part of an effort led by Sunni elder statesman Adnan Pachachi to ensure the community is included in the key next step after the vote.
"The representatives of these political bodies that did not participate in the elections have decided in principle to take part in the writing of the permanent constitution in a suitable way," a statement from the group said.
The groups were mainly small movements and it was not clear how representative they are. Pachachi, who ran for a seat in the National Assembly in last Sunday's balloting, brought the groups together.
Taqi said the Shiite-led Alliance had contacted Sunni parties. The Alliance wants to work with "those who withdrew and boycotted when it comes to forming the government and the drafting of the constitution," he said.
The 275-seat National Assembly will form the next government and draw up a permanent constitution, which will then be put to a national referendum.
In the latest violence, four Iraqi National Guardsmen died in a roadside bombing early Saturday in Basra, Iraq's second largest city, while gunmen overran a police station in the northern city of Mosul, killing five officers, police officials said.
Gunmen assassinated a member of the Baghdad city council, Abbas Hasan Waheed, and a member of Iraq's intelligence service in two separate drive-by shootings.
Two Iraqi troops died from a blast that hit a patrol in the central city of Samarra, and three Iraqi National Guard soldiers were also killed in clashes west of Mosul, in the city of Tal Afar, on Friday night, hospital officials said.
The two American soldiers from Task Force Danger were killed in a roadside bombing Friday night near Beiji, 155 miles north of Baghdad, the U.S. military said Saturday. The military had said a second roadside blast in the same area killed two more American soldiers, but later said that report was wrong.
West of the capital, a U.S. convoy in the insurgent stronghold of Ramadi was rocked by a roadside bomb that killed two Iraqi bystanders, a hospital official said. It wasn't clear if there were any American casualties.
The brother of the police chief for Mosul and the surrounding Ninevah province was kidnapped Saturday, police officials said.
The kidnapping came three days after the chief, Gen. Mohammed Ahmed al-Jubouri, had threatened to destroy rebel sanctuaries if insurgents did not surrender their weapons within two weeks.
Early Saturday, an Italian journalist received a call from the mobile telephone of colleague Giuliana Sgrena, 56, who was kidnapped by gunmen Friday near Baghdad University.
Radio journalist Barbara Schiavulli, who received the call from Sgrena's phone, heard only Arab music playing in the background, said Cristiana Tomei, a colleague of Schiavulli's speaking in Rome.
Also Saturday, an Iraqi police commander said 11 of his officers were missing after their convoy was ambushed this week in a western Baghdad suburb.
In the northern city of Mosul, the bodies of three unidentified Iraqis who had been shot in the head were found on the streets, police said. No other details were available.