Powell's trip is unlikely to end with a ceasefire
Mr Powell will hold a second round of talks with Palestinian leader Yasser Arafat on Wednesday, before flying to Cairo where US officials said he would brief Egyptian President Hosni Mubarak on his peace mission.
Jordanian Foreign Minister Marwan Moasher will also visit Mr Arafat in his besieged headquarters in Ramallah "in the next few days", the Jordanian Foreign Ministry said.
In Washington, the White House announced that Saudi Crown Prince Abdullah, who has proposed a regional peace plan, will meet President Bush in Texas on 25 April.
As the diplomatic moves continued, heavy fighting erupted around the Church of the Nativity in Bethlehem on Tuesday night, where about 200 Palestinian gunmen have been besieged by Israeli troops since 2 April.
Joint statement
The BBC's Jon Leyne in Jerusalem says it does not look like Colin Powell will achieve the ceasefire and full withdrawal of Israeli troops he had hoped for.
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Palestinian and US negotiators are reported to be working on a joint statement condemning suicide bombings and endorsing the creation of a Palestinian state, but falling short of declaring a ceasefire.
The Palestinians have demanded that Israel withdraw from Palestinian towns and villages in the West Bank before they sign a truce.
Israel continued its security operations on Tuesday, raiding three Palestinian suburbs of Jerusalem and the town of Tulkarm, from which they had previously withdrawn on 9 April.
The Israeli army said it had arrested four members of the Islamic militant group Hamas, before it withdrew from the town.
Palestinian witnesses said Israeli soldiers also swept into the Raffidiyeh area of Nablus and detained five people.
The raids were part of Israel's hunt for Palestinian militants, begun on 29 March following a spate of Palestinian suicide attacks on Israeli targets.
Bethlehem shoot-out
The BBC's Caroline Hawley, who is in Bethlehem, reported the heaviest exchange of fire between Israeli troops and Palestinian gunmen in the Church of the Nativity since the stand-off began two weeks ago.
Our correspondent said a bell tower had been hit and smoke was rising in Manger Square.
The governor of Bethlehem, Mohammed al-Madani, who is inside the church told Reuters news agency: "The Israelis have tried to storm a section of the church."
The Israeli army said it was operating in the area but denied firing on the church.
On Monday, Mr Sharon said Israeli forces would remain in Bethlehem and Ramallah until Palestinian "terrorists" there were handed over.
He said the army would, however, pull out of Nablus and Jenin within a week, a move welcomed by President Bush.