The Sciri is Iraq's largest Shia political party and is seen as key to Iraq
's ruling Shia alliance.
Although al-Hakim holds no official government post, he is thought to be one of the most powerful figures in Iraq.
The party's military wing, the Badr Brigade, is said to be 10,000 people 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 warned 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 comes just after a leaked White House memo, published in the New York Times, showed 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.
The White House played down the memo, calling it a "laundry list" of ideas, rather than a call for a new course of action, but Bush's meeting with al-Hakim, coming only a week after his meeting with Nuri al-Maliki, the Iraqi prime minister, has fuelled speculation that the White House is searching for additional sources of help in stabilising Iraq.
Joseph Biden, a Democratic senator and the incoming chairman of the Senate foreign relations committee, said: "The Rumsfeld memo makes it quite clear that one of the greatest concerns is the political fallout from changing course here in the United States. The bottom line is there is no one, including the former secretary, who thought the policy the president continues to pursue makes any sense."