White House talks with al-Hakim on Monday appeared to signal a more direct effort by Bush to stabilise Iraq - an essential to any eventual pullout of US troops.
The Supreme Council for the Islamic Revolution in Iraq (Sciri), was founded in Iran in 1982 and is part of the United Alliance, a grouping of Shia Islamist parties that won a near majority in the Iraqi parliament elected in December.
The US president told al-Hakim during the talks that he was not satisfied with the pace of progress in Iraq, and he later told Fox News: "What Americans are trying to figure out is why Iraqis are killing Iraqis when you have a better future ahead."
Although al-Hakim holds no official government post, he is thought to be one of the most powerful figures in Iraq.
Bush also told reporters that he and al-Hakim talked about the need for "elected leaders and society leaders to reject the extremists that are trying to stop the advance of this young democracy".
In a speech to the US Institute of Peace, which is co-ordinating work on the Baker group's report, al-Hakim criticised the US forces for not striking hard enough.
He said: "The strikes they are getting from the multinational forces are not hard enough to put an end to their acts.
"Eliminating the danger of civil war in Iraq could only be achieved through directing decisive strikes against Baathist terrorists in Iraq. Otherwise we'll continue to witness massacres being committed every now and then against innocent Iraqis."
The Sciri's military wing, the Badr Brigade, is said to be 10,000 strong. It was founded by al-Hakim in exile and is financed by Iran.
"The bottom line is there is no one, including the former secretary, who thought the policy the president continues to pursue makes any sense."
Joseph Biden Democratic senator
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The Badr Brigade is blamed for some of the sectarian violence in Iraq, although al-Hakim has said that the outbreak of a civil war in Iraq, pitting Sunnis against Shia, would spell disaster for Iraq and the region.
Al-Hakim's meeting with Bush came after a leaked White House memo, published in the New York Times, showed that Donald Rumsfeld, the former US defence secretary, admitted two days before he resigned that US strategy in Iraq was "not working well enough or fast enough" and "needed a major adjustment".
In the memo, Rumsfeld noted that the US mission had changed from "major combat operations, to counterterrorism, to counter-insurgency, to dealing with death squads and sectarian violence".
The document also outlined possible large reductions in troops in Iraq, in contrast to Rumsfeld's calls in public for the US to "stay the course", and a reduction of US bases in the country from 55 to five by July next year.
Bush praised al-Hakim for his "strong position against the murder of innocent life." But he also said: "I told him that we're not satisfied with the pace of progress in Iraq."
Possibly mindful of Washington's concerns about Sciri's ties to Shia Iran, al-Hakim told Bush that Iraq's neighbours should not become involved in the country's affairs.
Minority Sunnis have accused Sciri's fighters of carrying out assassinations.
The White House meeting with al-Hakim is seen by some analysts as a sign that Bush, whose Republicans suffered heavy election losses last month partly due to public disapproval over the conduct of the war, is delving deeper into the quest for a new strategy for Iraq.