Both sides say they're ready to make concessions as a goodwill gesture to President Bush, and the first of those moves could happen soon. Israel may release 100 Palestinian prisoners in the next few days, reports CBS News Correspondent Robert Berger, and has decided to ease a two-week long closure of the West Bank and Gaza Strip.
"We do want to move forward," said Israeli spokesman Mark Sopher. "We have accepted the vision of the American administration."
Other goodwill gestures include releasing millions of dollars in withheld tax revenues, and issuing work permits to 25,000 Palestinians.
However, later Friday, the U.S. Embassy warned on its Web site it had received "credible reports" of plans to kidnap U.S. citizens in Gaza.
"At this time, Americans are advised to be particularly cautious," the announcement said.
The announcement was followed by a statement urging U.S. citizens to defer travel to Israel, the West Bank and Gaza.
"Ongoing violence has caused numerous civilian deaths and injuries, including to some American tourists, students and residents," the announcement said. "The potential for further terrorist acts remains high."
Though statements from both sides early Friday, following Sharon's meeting with Palestinian Prime Minister Mahmoud Abbas, were couched in positive terms, in the end, they maintained their basic positions.
The Palestinians demanded implementation of the U.S.-backed "road map" plan leading to a Palestinian state, and the Israelis conditioned all additional steps on a crackdown against violent groups responsible for attacks against Israelis.
"That is stated openly black and white in the road map itself as part one, phase one, paragraph one," said Sopher.
Abbas prefers stopping violence through dialogue and told an Israeli newspaper that an agreement with the largest militant group, Hamas, could be completed next week. Hamas has been responsible for most of the suicide bombing attacks that have killed more than 300 Israelis in 32 months of violence.
The three-hour meeting Thursday night between Sharon and Abbas, their second in two weeks, came ahead of a summit between the two leaders and Mr. Bush planned for the Jordanian resort of Aqaba next Wednesday.
Neither side wants to disappoint the U.S.
"Nobody can ignore President Bush or marginalize the importance of America," said Palestinian Foreign Minister Nabil Shaath.
Before that summit, Mr. Bush will meet with Arab leaders in Cairo to enlist their support. In an interview with the Dubai-based Arabic satellite channel, Al Arabiya, the president said he would tell Arab leaders that peace is possible and that he will work to achieve that goal. President Bush's remarks were simultaneously translated into Arabic, and an English transcript was not available.
The Palestinians are demanding Israel make a clear statement endorsing their right to a state at that meeting.
Declarations by each side recognizing the rights of the other to statehood and security are supposed to be the first step of the road map, which begins with a halt to 32 months of bloody violence and leads to a full-fledged Palestinian state in 2005.
The Israelis are insisting that first, the Palestinians must dismantle armed cells, confiscate illegal weapons and jail militants.
Two of the prisoners the Israelis may release, at the request of the Palestinians, are Khaled Abu Sukar, the oldest Palestinian held by the Israelis, said to be in his 50s, and Tayseer Khaled, a member of the Palestinian Liberation Organization's executive committee.
Israel repeated its offer to pull its troops back from West Bank Palestinian towns and cities reoccupied in a series of incursions over the past year and redeploy troops in Gaza to give Palestinians a chance to enforce security, an Israeli term for preventing terror attacks.
However, Shaath said this could be done only "after reaching an agreement on a cease fire with Hamas and the other Palestinian groups. Now we will try our best to reach this agreement ... and I'm optimistic."
Moving in the direction of the declaration the Palestinians demand, the Israeli statement said that if the Palestinians eliminate terrorism, Israel would negotiate a peace agreement according to principles set down in a Bush speech from last June, "in which framework it will be agreed to establish a provisional Palestinian state and afterward a permanent state."
Israel's government prefers to link its policy to the Bush speech, rather than the road map, because it objects to some of the wording in the international plan, sponsored by the European Union, Russia and the United Nations along with the United States.
After the meeting, the Palestinian participants returned to the West Bank town of Ramallah to brief Palestinian leader Yasser Arafat. He said he would convene a meeting of the PLO executive to review the results of the talks.