On Friday, Shia leaders met but failed to arrive at a consensus on naming a successor.
Al-Jaafari's party, Dawa, is backing Jawad al-Maliki as the new nominee, according to three officials from different Shia factions. All spoke on condition of anonymity because of the senstivity of the talks.
However, the biggest Shia party, the Supreme Council for the Islamic Revolution in Iraq, opposes al-Maliki because it fears he will be unacceptable to Sunni Arabs. Instead, the supreme council, or SCIRI, backs Ali al-Adeeb, who is also a Dawa member.
On Friday, another party in the Shia coalition, Fadhila, offered its own candidate - Nadim al-Jabiri - for the prime minister's job, party officials announced in Najaf.
Members of parliament said Grand Ayatollah Ali al-Sistani, Iraq's most influential Shia cleric, also favours al-Adeeb, believing he would be acceptable to Sunnis and Kurds.
Al-Sistani was instrumental in pressuring al-Jaafari to agree to allow Shia politicians to choose a new nominee after resisting pressure from Sunnis and Kurds to step aside.
The Shia are the biggest bloc in parliament but lack the strength to govern without Sunni and Kurdish partners. As the biggest bloc, the Shia get first crack at the prime minister's job.
But the removal of al-Jaafari threatens to create new problems because of the lack of alternative candidates with the stature and power to confront the nation's problems. Each has a record which could cause discomfort among the major players.
For example, al-Adeeb was a member of the political bureau of Dawa based in Tehran in the 1980s - a time when Western governments considered the party little more than an instrument of the Iranian intelligence service.
Al-Maliki was deputy chairman of the committee charged with purging officials of Saddam Hussein's Baath party from government service and politics. Many Sunnis believe the committee's goal was to strip Sunnis of their rights.
The heads of the seven factions of the Shia alliance were to meet on Friday evening to see if they could resolve the differences. If not, the names of the candidates could be put to a vote among the 130 alliance members of parliament.
Shia officials hope to resolve the issue before a planned meeting of the Iraqi parliament on Saturday afternoon. But some officials were pessimistic, and one said the split could develop into a new crisis within the alliance.
Sunni and Kurdish opposition to al-Jaafari produced a deadlock in efforts to form a new government. On Thursday, al-Jaafari bowed to pressure and agreed to let Shia MPs choose a new nominee.
US officials are insisting that the Iraqis move quickly to form a new government to begin the task of confronting sectarian violence and armed insurgency.