Followers of Moqtada al-Sadr, an influential Shia Muslim leader, are holding a poll to establish who has the people's support to become the next prime minister.
The movement said the two-day "referendum", starting on Friday, would give all Iraqis a chance to choose which of five candidates they would favour as prime minister.
The decision to hold the vote was made by Sadr himself, because "the political parties that contested the election have been unable to select a prime minister," Ahmed Khalaf, one of the organisers of the poll, said.
"The parties are neck-and-neck and they all have a candidate for the post ... and this has prevented the rapid formation of a government."
Leaders of Iraq's various political groups have been involved in discussion aimed at finding common ground with none of the four major blocs close to forming a majority on their own.
Leaders on the list
Final election results gave the Iraqiya coalition led by Iyad Allawi, a former prime minister, just two seats more than State of Law bloc led by Nouri al-Maliki, the incumbent prime minister.
Both al-Allawi and al-Maliki are on the Sadr movement's ballot, along with al-Maliki's predecessor Ibrahim al-Jaafari, Adel Abdel Mahdi, a vice-president, and Jaafar al-Sadr, who ran on al-Maliki's list but has al-Sadr family ties.
Hazem al-Araji, a Sadrist leader, said the referendum, to be held nationwide, will not be a repetition of the March 7 election.
"Iraqis chose who they wanted as their MP and not their prime minister," he said, referring to the general election.
The Sadrist voting, which has no legal authority, was held at al-Sadr offices, mosques and other sites across the country.
"People coming here are happy, they are treating it like a festival because they are practising their right to choose their prime minister," Saleh Jizani, the head of a voting centre in north Baghdad, said.
Sadrist officials were seen carrying ballot boxes around Baghdad, stopping people on the street and arriving on locals' doorsteps to ask them to vote.
Al-Sadr's followers won at least 39 seats in the 325-seat parliament in the election, making them the largest group within the Iraqi National Alliance (INA), a Shia coalition that placed third in the race.
"With this referendum, the Sadrists have made a shrewd move to put pressure on the other political parties"
Hamid Fadhel, political science professor at Baghdad univeristy
Sadrist support is crucial for al-Maliki as he tries to assemble enough Shia backing to remain prime minister.
But the enmity between the Sadrists and al-Maliki is deep-rooted and analysts say al-Sadr, who has been living in Iran for the last two years, may use the poll as a tool to push him from power.
If another candidate comes out on top in the Sadrist poll, the influential Shia figure would be able to throw his support behind someone other than al-Maliki.
"With this referendum, the Sadrists have made a shrewd move to put pressure on the other political parties," Hamid Fadhel, a Baghdad University political science professor, said.
"The negotiations with State of Law have stalled and the Sadrists want to push for someone other than Nouri al-Maliki [as premier], armed with popular support.
The Sadrists were key supporters when al-Maliki formed his government in 2006.
But two years later, he turned the security forces on the Mahdi Army, the movement's armed wing, jailing thousands of al-Sadr supporters in a campaign to destroy militias in the southern city of Basra and the Baghdad suburb of Sadr City.
Meanwhile, Allawi received a major boost to his hopes of becoming prime minister again after the leader of a key faction within the INA said it would not join a government without him.
"We will not participate in a government that does not include Iraqiya," Ammar al-Hakim of the Supreme Iraqi Islamic Council said on Thursday.
"It received many votes in the western regions and in Baghdad, and it is not right to ignore the will of these people, because excluding Iraqiya means excluding these people."