King Abdullah plans to use his time with Bush to urge him to find a resolution to the Arab-Israeli conflict which he says is the root cause of the violence in the Middle East.
However the main focus of the discussions will be the need to find a solution to the increasingly heavy sectarian fighting in Iraq.
On Tuesday, Bush said he would not pull US troops out of Iraq until their mission had been completed and sidestepped questions over whether there was now civil war in the country.
Bush said in a speech on his arrival in the Latvian capital Riga for a Nato summit: "One thing I won't do, I am not going to pull the troops off the battlefield before the mission is complete."
However, ABC television on Tuesday reported Pentagon officials as saying the US military was considering withdrawing its soldiers from Anbar province in western Iraq.
US forces have suffered high casualty rates in heavy fighting in the province.
General Peter Pace, the chairman of the Joints Chiefs of Staff, was quoted on the ABC website as saying the US "is considering turning Anbar over to Iraqi security forces and moving US troops from there into Baghdad".
ABC said another option, opposed by senior military figures, was to increase the number of soldiers in Iraq for a short-term period. There are currently about 139,000 US troops in Iraq.
A Pentagon spokeswoman could not confirm the report.
Iraq panel debate
Meanwhile, the Iraq Study Group, an independent commission on Iraq policy, could not reach a consensus on how many or how long US soldiers should remain in Iraq.
The group will meet for a third day of debate on Wednesday.
Members made little comment as they left the Woodrow Wilson Centre on Tuesday afternoon. Leon Panetta, former White House chief of staff to Bill Clinton, said: "Trying, ... trying to find consensus... It goes up and down."
Iraqi forces may be asked to take over
Anbar province from the US military
The panel is led by James A Baker III, a former Republican secretary of state, and Lee Hamilton, a former Democratic House member.
They are having difficulties reaching agreement on what the appropriate level of US soldiers should be in Iraq, whether there should be a phased withdrawal, and if so, under what time-frame, an official, who spoke on condition of anonymity because the panel's deliberations are private, said.
A second official said the commission was unlikely to propose a timetable for withdrawing US soldiers from Iraq but that some members seemed to favour setting a date for only an initial withdrawal, an idea that has been pushed by many congressional Democrats.
The group was initially expected to wrap up their deliberations on Tuesday. People close to the commission said they expect the panel to issue a bipartisan, consensus report some time next month.
Officials have said a consensus report is the group's goal, rather than having the panel issue one that suggests two divergent opinions with no clear recommended path.