The US-appointed Council has delayed until Tuesday a meeting to nominate a new president to take over sovereignty from the US-led occupation force, Kurdish member Mahmud Uthman has said.
The two main candidates in the race to succeed Saddam Hussein are running neck and neck.
Tribal magnate Ghazi al-Yawir and elder statesman Adnan Pachachi are both moderate Sunni Muslims and Iraqi exiles with strong ties to Washington.
"I think we are going to win either way. They are actually both very forward looking, moderate gentlemen," one official from the US-led occupation told AFP.
Strange then, that there should be so much haggling over men with similar attributes for a job stripped of its Saddam-era powers and whittled down to a mainly ceremonial role.
A tribal magnate and businessman, at only 46, al-Yawir is distinctive for his rather chubby face and characteristic smile. But his traditional flowing white tribal dress and kuffiya belie his Western connections and education.
After studying engineering at George Washington University, he moved to Saudi Arabia where he opened a prosperous telecommunications business.
Saddam's demise changed his plans.
Shaikh Ghazi al-Yawir has lived
for 15 years in Saudi Arabia
He returned to Iraq at the behest of his uncle after Saddam fell and became a member of the Governing Council, of which he was appointed president after his predecessor Izz al-Din Salim was killed in a bombing on 17 May.
From the multi-ethnic northern city of Mosul, al-Yawir has promoted his inter-sectarian ties, saying he has close relations to Kurds and that his mother taught him to respect the Shia as well as Sunni tradition and Christianity.
The Shammari tribe, of which he is a leader, is one of Iraq's largest and incorporates about three million Sunni and Shia.
During fierce fighting in April in the flashpoint Sunni town of Falluja, both al-Yawir and Pachachi slammed a US military offensive in which hundreds died despite the campaign's mission of hunting down the killers of four US contractors in March.
Both men appealed for an end to the bloodshed and tried to negotiate an end to the conflict.
At 81, Pachachi is nearly twice Yawir's age and draws on a wealth of experience as the scion of a long-established political family and a leading light in the world of Iraqi expatriates forced abroad under Saddam.
His father, uncle and father-in-law were all prime ministers during the time of the British-installed monarchy after World War One.
Adnan Pachachi is 81, nearly
twice the age of his rival al-Yawir
Before being forced into exile in 1971, three years after the Baathist coup, the secular Sunni was Iraqi representative to the United Nations and foreign minister from 1966 to 1967.
Also educated in Washington, at the elite Georgetown University, where he took a PhD in political science, Pachachi has always championed his independence, despite his close ties to the US State Department.
As Washington moved towards its invasion of Iraq in the early months of 2003, Pachachi made a swift return to political life, after announcing his retirement in the late 1990s.
Even before tanks rolled across the border, Pachachi called for the United Nations to take a prominent role in handling the country once Saddam was removed and warned against a lengthy occupation period.
He returned to Baghdad last year and was given a seat on the Governing Council, of which he was president for Janaury.