Allawi told a press conference in a Damascus hotel that he and Sadr had agreed on "the need to speed up the formation of a government that would group all the political movements and have a clear programme."
Separately, Sadr told reporters that Allawi's bloc was "ready to make concessions to put an end to Iraq's political crisis," and urged other parties to do the same "for the sake of the Iraqi public interest."
Neighbouring countries are welcome "to give advice" but "without interfering in Iraqi affairs," he said.
In his talks with Allawi, Assad reiterated "Syria's support for any inter-Iraqi accord (on a government) which conserves the unity of Iraq, its Arab identity and its sovereignty," Syria's official news agency SANA reported.
Allawi, who is vying for the post of prime minister with the incumbent, Nuri al-Maliki, in turn thanked Syria for playing host to hundreds of thousands of Iraqi refugees and its support for efforts to restore stability in Iraq.
Efforts to form a new government, more than four months after a March 7 election in Iraq, also figured in talks in Damascus last Saturday between Assad and Sadr.
The bloc of anti-US cleric Sadr, who lives in self-imposed exile in Iran, gained 39 seats in Iraq's new 325-strong parliament, against 91 for Allawi and 89 for Maliki -- both also Shiites.
On Monday, Turkish Foreign Minister Ahmet Davutoglu flew in to Damascus and held talks with Assad, agreeing on the need "to speed up the formation of a government" in Iraq, their two countries' common neighbour, SANA said.
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