Gaza Strip ? Egyptian mediators asked Palestinian militants Sunday to halt attacks on Israel in an effort to rescue a U.S.-backed peace plan that foundered after more than a week of violence. Israel’s prime minister suggested he might accept a limited truce.

Palestinian officials said there were expectations that the current cease-fire efforts might succeed where others had failed, despite militia leaders’ renewed assertions they will not lay down their arms.

In Maine, President Bush said he was confident of eventual peace between the longtime enemies, with a Palestinian state existing in peace beside Israel.

After a weekend meeting between Israeli and Palestinian security officials, both sides said Israel was considering a gradual withdrawal from Bethlehem and parts of the Gaza Strip, leaving security to the forces of Palestinian Prime Minister Mahmoud Abbas and security chief Mohammed Dahlan.

Addressing a Cabinet meeting Sunday, Israeli Prime Minister Ariel Sharon said that if militants agree to a cease-fire, Israel would, for the most part, respect it.

“If no one fires on us, we will not return fire, except in cases of ticking bombs,” a Cabinet official quoted Sharon as saying.

Until now, Sharon had rejected the truce idea out of hand, demanding an immediate crackdown on the militants, including arresting their leaders.

However, Sharon’s new comments appeared highly conditional. Until recently, the phrase “ticking bomb” defined a militant about to carry out an attack. Then the Israelis widened the definition after a botched effort to assassinate Hamas leader Abdel Aziz Rantisi last week, calling him a “ticking bomb.” That indicated that even if Hamas agrees to a truce, its leaders might not be safe from Israeli strikes that could rekindle violence.

In a reflection of the unpredictable dynamic of the Mideast conflict, other hopeful signs emerged over the weekend for the so-called “road map” plan, which envisions an end to 32 months of violence and the creation of a Palestinian state by 2005.

Israeli police officers drag away Israeli youths who tried to prevent the evacuation of the illegal outpost of Asael, south of the West bank town of Hebron. The illegal outpost is likely to be demolished by the Israeli army under the terms of the U.S-backed road

In talks Saturday with Israeli Maj. Gen. Amos Gilad, Dahlan proposed that Israeli troops withdraw from large areas of Gaza to positions held before the outbreak of fighting in September 2000. He asked Israel to pull out of the West Bank towns of Bethlehem and Ramallah, the Palestinians’ administrative center.

Israeli Defense Minister Shaul Mofaz said Sunday he is considering a withdrawal from Bethlehem and parts of Gaza and said meetings with the Palestinians would continue.

In a first phase, Israel would pull out of the northern Gaza towns of Beit Hanoun and Beit Lahiya. Despite the talks, Israeli forces entered Beit Hanoun early Sunday, setting off a firefight that killed one Palestinian and wounded seven, doctors said.

And U.S. officials, underscoring their nation’s commitment to the peace plan, arrived in the region to monitor its implementation.

The two Egyptian mediators, assistants to Egyptian intelligence chief Omar Suleiman, arrived in Gaza City on Sunday for cease-fire talks with Palestinian militants.