Israel says it reserves the right to deploy its army
The Israelis plan to pull troops and tanks out of Bethlehem and its suburb of Beit Jala in the evening.
Raanan Gissin, an advisor to Israeli Prime Minister Ariel Sharon, said the pull-out would serve as a "test case" for similar withdrawals, which would require "complete quiet" in the area.
But the Palestinians say calm can only begin after Israel pulls out.
"Our position is that calm could only begin by Israeli withdrawal from our areas," said Jibril Rajoub, the chief of Palestinian preventive security in the West Bank.
Palestinian officials said the parties would meet on Sunday to discuss further Israeli redeployments.
Quiet night
A BBC correspondent in Jerusalem says Friday night was the quietest night in the West Bank since the Israelis went into six Palestinian areas last week, following the killing of the Israeli tourism minister, Rehavam Zeevi.
More than 40 Palestinians have been killed during the Israeli incursions, many of them in the village of Beit Rima.
The United States, which chaired the meeting between the Israelis and Palestinians, has led international demands for Israel to pull out.
White House spokesman Ari Fleischer said President Bush was pleased with the first step of the withdrawal, but said the president expected Israel to complete it.
He said the president also called on Palestinian leader Yasser Arafat to "make a 100% effort to reduce the violence," and for the sides to resume peace talks.
Before Friday's meeting, the United Nations Security Council issued a statement calling for the "immediate withdrawal of all Israeli forces" from Palestinian areas.
Friday violence
On Friday, Israeli troops shot dead three armed Palestinians who the army said were trying to infiltrate a Jewish settlement.
The militant Palestinian group Hamas said the men were its fighters and had attacked the Israelis with a mortar, machine guns and grenades.
Israeli soldiers also shot and killed an Israeli Bedouin Arab by the fence between Israel and Gaza. The army said he ignored orders to stop as he drove a tractor through the fence.
The Israeli army initially said it planned to remain in the reoccupied areas until the Palestinians handed over the militants responsible for the assassination of the Israeli tourist minister.
That has not happened. By Israel's own admission, two of those responsible for the murder remain at large.
The BBC's Stephen Gibbs in Jerusalem says that even before it has happened, the permanence of any Israeli withdrawal remains open to question.
The Israelis have repeated that they reserve the right to deploy their army as they see fit, according to their security needs.