Hakim in February was said to support Allawi for the position of prime minister over the incumbent, Nouri al-Maliki.
Hakim leads the Shiite political movement Supreme Islamic Iraqi Council. His group had joined with Maliki's State of Law party, however, after a March 7 vote to form a grand Shiite coalition that was four seats shy of the majority needed to form a new government.
Divisions over the top positions in the Iraqi government, however, have resulted in a six-month stalemate in forming the new government in Baghdad.
Hani Ashour, an adviser to Iraqiya, told the Voices of Iraq news agency that his bloc intends to present a draft resolution to Hakim's political slate with the aim of forming the new government.
Hakim's political movement is part of the Iraqi National Alliance, which secured 70 seats in the March elections. Iraqiya won 90 seats but was well short of the 163-seat majority needed to form a government alone.