Updated

U.S. Secretary of State Condoleezza Rice's shuttle diplomacy brought her to the Middle East for consultations with Israeli and Lebanese leaders on U.S. proposals intended to end the violence.

Rice stressed the difficulty of ending the three weeks of fighting between Israelis and Hezbollah guerrillas who dominate the border region. Yet the chief U.S. diplomat is hoping to make strides toward an agreement that will clear the way for a U.N. resolution — complete with peacekeepers — within a week.

"These are really hard and emotional decisions for both sides, under extreme pressure in a difficult set of circumstances," Rice told reporters Saturday on her trip from Malaysia, where she attended a meeting on Asian issues. "And so I expect the discussions to be difficult but there will have to be give-and-take."

Rice had dinner with Israeli Prime Minister Ehud Olmert after arriving in Jerusalem, but did not speak publicly afterward.

Israeli officials said Rice and Olmert agreed that a resolution of the conflict must be based on a previous U.N. Security Council resolution calling for disarming Hezbollah and deploying the Lebanese army on the border instead.

The officials, speaking on condition of anonymity because they were not authorized to talk to reporters, said Olmert and Rice also agreed that the conflict must not end with restoration of the situation in which Hezbollah could fire rockets at Israel at will.

They said Rice emphasized the humanitarian issues facing Lebanon, and Olmert pledged cooperation. The Israeli prime minister noted that more than 100,000 Israelis are confined to cramped bomb shelters because of the Hezbollah rocket attacks.

On Sunday, she planned to meet with Israeli Foreign Minister Tzipi Livni and Defense Minister Amir Peretz.

Officials have not disclosed who else she might see during her second stopover in a week to the Middle East. Rice's weeklong trip has included last-minute schedule changes, including a surprise visit to Beirut last Monday.

Israel Radio reported Rice plans to travel to the United Nations on Tuesday and hopes the Security Council can prepare a resolution calling for a cease-fire on Wednesday. The report said Israel would not agree to stand down until Hezbollah was incapable of resuming rocket fire at Israel.

All sides seemingly want to end the fighting. But the United States and Israel are pressing for a settlement that addresses enduring issues between Lebanon and Israel, rather than an immediate cease-fire favored by numerous world leaders.

The trick will be negotiating a plan — and a sequence of events that satisfies both sides — to address contentious issues such as prisoner exchanges and border disputes.

A small piece of land, Chebaa Farms, could prove critical to a resolution. Israel has occupied it since 1967, despite Lebanese claims.

Hundreds have died in the fighting, mostly Lebanese civilians, drawing international condemnation. Less than four hours after Rice's arrival, a U.N. spokesman said two U.N. peacekeepers were wounded by an Israeli airstrike that hit their post in southern Lebanon. Last week, four peacekeepers died in an Israeli airstrike.

A top U.N. peacekeeping official said Saturday he feared the war in southern Lebanon would continue until late August.

Rice's diplomatic effort did not impress Hezbollah's leader. In a televised address, Sheik Hassan Nasrallah said the Bush administration wants the fighting to continue. He threatened more attacks deep into Israel "if the savage aggression continues on our country, people and villages."

On her flight to the Middle East, Rice told reporters she was encouraged by a proposal drafted by the Lebanese Cabinet to end the fighting with Israel. The cabinet includes two members who support Hezbollah, and she said their agreement — in principle — is evidence the fragile democracy can work like a healthy government.

Yet it was unclear if Hezbollah would follow through on the plan, which effectively could abolish the organization's military wing. "The most important thing," Rice said, "that this does for the process is that it shows a Lebanese government that is functioning as a Lebanese government."

Fighting between Hezbollah's guerillas and Israel exploded after Hezbollah's July 12 raid into Israeli territory. Two Israeli soldiers were captured. Israel says their release is pivotal to ending the current crisis. Rice, too, renewed her call for their release.

The secretary said she would not set deadlines for reaching any international agreements nor say how close Israel is to achieving its strategic goals for dismantling Hezbollah, its leading justification for the fighting.

Rice has not offered details publicly on what the international force should look like. Privately, numbers discussed for its size range from 15,000 to 20,000 troops.

Without offering a time line, Rice said countries with "ready forces" that have their own logistical support could go quickly into Lebanon. Various states could be asked to contribute forces, which would operate under a U.N. mandate.

She said a U.N. force often takes more time to assemble and deploy.

France, Italy, Germany, Ireland and Turkey have said they are considering joining a U.N.-run multinational force.