Most jobs are in the military, police, and intelligence forces. But Iraqis say those jobs are only attained through family ties or bribes.
By the end of July, they hope to finalize a deal that would map out the role and "time horizon" for US troops in the country.
The Islamic Supreme Council of Iraq aims to capitalize on the disarray within Moqtada al-Sadr's movement ahead of provincial elections planned for October.
A Tuesday bomb attack at a municipal council building within a fortified section of Baghdad's Shiite enclave killed at least four Americans and six Iraqis.
On Sunday, Iraqis focused on their country's struggle on a soccer field in Dubai instead of the war on the streets.
Iraqi troops took the southern city without a shot being fired from Shiite cleric Moqtada al-Sadr's Mahdi Army.
The military blamed a little-known Shiite cell leader for Tuesday's attack in the Shiite neighborhood of Hurriyah that killed at least 63 people.
A Mahdi Army truce holds most of the Shiite cleric's forces in check. But some terrorize residents of Risala, a Baghdad neighborhood.