In addition to meeting Kuwaiti leaders, Rice is also due to meet foreign ministers of the six Gulf Co-operation Council (GCC) member states plus Egypt and Jordan in a bid to rally support for Bush's "surge" strategy to tackle violence in Iraq with the deployment of an additional 21,500 troops.
The GCC includes Bahrain, Kuwait, Oman, Qatar, Saudi Arabia and the United Arab Emirates.
The plan, revealed last week, has come under fire in many Arab capitals, even among staunch allies in the Gulf, with critics saying it provides a recipe for more sectarian violence in Iraq that could spread elsewhere in the region.
But Rice on Monday picked up support from Egypt after meeting with Hosni Mubarak, the president, in the southern city of Luxor.
"We are supportive of the plan ... We are hopeful that plan will lead to the stabilisation of, unity and cohesion of the Iraqi government," said Ahmed Abul Gheit, the Egyptian foreign minister.
While Rice began her trip stressing that she had no "plan" for reviving Middle East talks, she announced a three-way summit with Ehud Olmert, the Israeli prime minister, and Mahmud Abbas, the Palestinian president.
The summit, expected to take place in three or four weeks, will be the first meeting between Israeli and Palestinian leaders in two years. Stressing the meeting would only be a prelude to the resumption of final settlement negotiations.
Rice said: "There are a number of issues, some old, some new, that will have to be resolved if there is to be a Palestinian state. "I am very clear about one thing we do not want to do, which is to rush the formal negotiations before things are fully prepared, before people are fully prepared."
Olmert welcomed the summit but stressed that any Palestinian government involved in peace talks should recognise Israel's right to exist.
In the West Bank town of Ramallah on Sunday, Abbas said he rejects: "Any temporary or transitional solutions, including a state with temporary borders, because we do not believe it to be a realistic choice that can be built upon."
Abul Gheit said the Palestinian political crisis needed to be contained and contacts resumed between Israel and the Palestinians, in what he described as "a stabilisation phase".
He said: "Then you will start the second phase whereby everything is discussed in relation to the establishment of this Palestinian state."
During Rice's brief visit to Jordan late on Sunday, King Abdullah II told her that concrete progress needed to be made on the peace blueprint if the region was to be spared fresh bloodshed.
"Without tangible, specific steps to activate the implementation of the roadmap in the near future, the cycle of violence will widen," he said.
But Rice's diplomatic offensive suffered a blow earlier on Monday when Israel invited bids for new Jewish settlement homes in the West Bank even as she was exploring solutions for the region in a meeting with Olmert in Israel.
Settlements on occupied Palestinian territories are illegal under both Israeli and international law.