In an attempt to end the escalating violence, Bush said in a speech Thursday that he would send Secretary of State Colin Powell to the region next week.
Fighting continued in the West Bank, as Israeli troops took over Nablus, fought intense battles with gunmen barricaded in refugee camps and tightened a cordon around armed Palestinians holed up in Bethlehem's Church of the Nativity.
Earlier this week, Sharon had turned down a request by Powell to grant permission to U.S. mediator Anthony Zinni to hold talks with Arafat. For a week, the Palestinian leader has been confined to a few rooms in his headquarters in the West Bank town of Ramallah, and Sharon has said he is determined to keep Arafat isolated.
Sharon changed his mind after meeting with Zinni on Thursday, his office said in a statement issued after Bush's speech.
A senior Bush administration official, speaking on condition of anonymity, said Zinni would try to meet with Arafat on Friday.
With Powell standing at his side in the White House Rose Garden, Bush urged Israel to halt its incursions in the West Bank, launched last Friday after a series of suicide attacks on its citizens. Bush said that "storms of violence cannot go on," but also that the Palestinians must stop the suicide bombings.
Blowing yourself up does not help the Palestinian cause," he said.
Israel's government, which had hoped for more time to complete the operation, had no immediate reaction. Raanan Gissin, spokesman for Sharon, said only, "We have to study it and then we will have a statement."
But Finance Minister Silvan Shalom, an ally of Sharon's, was cautiously welcoming, saying that "if there will be a cease-fire, we'll be happy to pull out."
If they (Palestinians) come and say that they are stopping the fire and are stopping the attacks, there's no reason why we should be inside," Shalom said.
Ahmed Qureia, the speaker of the Palestinian Parliament, said Bush's comments were "new and important" and he praised the U.S. leader for urging an end to the offensive.
"We hope that Secretary Powell will bring with him a mechanism to implement this American vision," he said.
Meanwhile, the violence continued.
The Israeli military prevented reporters from reaching the church to assess the rival claims. All six West Bank towns taken over by Israel in the past week have been declared closed military areas, and reporters have been ordered to leave.
Five Palestinians were killed in Thursday's fighting, including three gunmen and a church caretaker, who witnesses said was shot while walking to the Church of the Nativity from his home.
World concern for the explosive situation was growing. A senior European diplomat said U.S. mediation efforts had failed, and that Washington should step aside as primary peacemaker.
As Bush spoke, Egyptian President Hosni Mubarak said in an address to his nation that Israel's military campaign will create hatred among 300 million Arabs. He urged the Bush administration to "exert its maximum effort" to ensure an Israeli withdrawal from the Palestinian territories.
Earlier Thursday, Israel blocked a high-level European mediation attempt, saying European Union foreign policy chief Javier Solana and Spanish Foreign Minister Josep Pique would not be permitted to meet with Arafat, who has been in what amounts to Israeli custody for a week. The European diplomats met Thursday with Israeli Foreign Minister Shimon Peres.
Israel launched "Operation Defensive Shield" last Friday to crush Palestinian militias that have carried out deadly attacks on Israeli civilians, now including seven suicide bombings in the past week. The trigger was an attack at the start of the Passover holiday that killed 26 Israelis attending a Seder, a ritual meal, in a hotel.
Since then, Israeli forces have taken over six major West Bank towns and cities - Ramallah, Qalqiliya, Jenin, Tulkarem, Bethlehem, Nablus - and have arrested more than 1,100 Palestinians. The towns of Jericho and Hebron remained the last islands of Palestinian control in the West Bank. The incursion into Nablus began late Wednesday, with dozens of tanks rolling into the city of 180,000, the largest in the West Bank. Gunmen took refuge in Nablus's Casbah, or old city, and in four adjacent refugee camps, where alleys are too narrow to allow tanks to enter.
A Palestinian man was killed by Israeli fire when he opened a window in his apartment in Nablus, Palestinian security officials said. Amar Yassin, a resident of downtown Nablus, said more than 80 armored vehicles were deployed in and around the main square and soldiers took over high-rise buildings.
The heaviest fighting raged in the Jenin refugee camp, a militant stronghold where hundreds of gunmen were holed up. Israeli commandos moved house-to-house, under fire cover from helicopters and tanks. At a nearby army command center, officers had an aerial photo of the camp pinned on the inside of a canvas tent. By Thursday morning, about 30 percent of the homes had been marked as having been searched or taken over by troops.
Over walkie-talkies, soldiers in the camp called for helicopter fire on a particular house. The commander of the operation, Brig. Gen. Eyal Shlein, said he would consider the raid a success when gunmen had been killed or taken captive.
"When I get to every single spot in the refugee camp ... and we've killed a few, that's how I will know (it was a success)," Shlein said. "We are not leaving any escape routes."
Palestinians in the camp said three gunmen were killed in the fighting, and that armed men had surrounded two buildings taken over by Israeli troops.
Gunmen in the camp said they believed this was their last stand, judging by the tough army sweeps through other West Bank towns. Armed men had prepared large numbers of homemade bombs for the Israeli raid of Jenin, the seventh in 18 months of fighting. Ali Safouri, a militia leader, said he and his men were trying to make every bullet count. "We use it for sniping only, we are not shooting in all directions," Safouri said in a telephone interview.
The standoff at Bethlehem's Church of the Nativity began Tuesday, the day Israeli forces seized the biblical town. After several hours of heavy fighting around Manger Square, dozens of gunmen ran into the basilica and, according to witnesses, forced their way inside, where they remain along with several dozen clergy.
A video of the incident, released Thursday by the Israeli military, showed the gunmen, a dozen at a time, running from the nearby Palace Hotel into the church, their heavy footfalls splashing puddles under slashing rain. "One at a time," shouted one of the men. Wearing military vests and boots and carrying rifles, each ran as another turned and provided cover, wildly shooting an assault rifle. Israeli Defense Minister Binyamin Ben-Eliezer said troops would not storm the church, but army officials said the gunmen would not be allowed to go free.
On Thursday, Palestinians inside the church said Israeli troops blew open a back door leading into a small courtyard adjacent to the church. Mazen Hassan, a Palestinian policeman, said soldiers fired into the courtyard, wounding three people.
Lt. Col. Olivier Rafowicz, an Israeli army spokesman, denied soldiers were in the church compound. Other military officials said there was shooting in nearby Manger Square, and that troops were pursuing gunmen.
Rafowicz said Israel has been offering safe passage out of the church for anyone wishing it, and that Palestinian officials holed up inside "are preventing the people from leaving."