UNITED NATIONS — Vice President Cheney will travel to the Middle East on Friday to talk peace and coordinate strategy on “regional threats,” President Bush said yesterday, announcing the trip as reports emerged of a possible Egyptian-brokered cease-fire between Israel and Hamas.

While officials in Gaza and Jerusalem denied that any such agreement had been reached, Israeli and Palestinian Arab analysts say both sides in conflict, as well as Egypt, have reasons for dialing back the intensity of the fierce fighting that has threatened to mature into a full-fledged war over Gaza.

A long stay in Gaza would monopolize the Israeli army’s assets at a time when unforeseen regional pressures could require urgent attention, according to Israeli analyses. An Israeli news report last night, citing intelligence sources, said the fighting in Gaza has not fully died down and tensions along the Lebanese border could soon flare up again. Hezbollah has “completed its military and logistical preparations” for a renewed confrontation, the report on the Ynet Web site said.

Mr. Cheney’s visit the Middle East is aimed at consolidating “strategic and regional” issues, including a coordinated strategy on Iran and Lebanon, rather than dealing with day-to-day post-Annapolis peace negotiations, a Washington-based official who requested anonymity said.

The vice president’s goal is to “reassure people” in the region that the administration is committed to a “vision of peace,” Mr. Bush said at the White House during a visit yesterday with Prime Minister Tusk of Poland. But the trip also will reinforce “that we fully see the threats facing the Middle East — one such threat is Iran — and that we will continue to bolster our security agreements and relationships with our friends and allies,” the president added.

The administration’s envoy to the region, General William Fraser, is expected to begin consultations with Israeli and Palestinian Authority officials later this week, and he is more likely than Mr. Cheney to touch on issues like Israel’s decision to expand construction in East Jerusalem or a possible agreement on Gaza, officials in Washington and Israel said.

“We all have an interest in calm on the ground,” a spokesman for Mr. Olmert, Mark Regev, told The New York Sun yesterday, referring to the situation in Gaza. “But we do not want an artificial calm that would only be exploited by Hamas to regroup, rearm, and prepare for the next round.”

If rocket attacks from Gaza “continue, we will shoot. If it will not, we will have no reason to shoot,” Mr. Olmert said yesterday in Jerusalem. He added that beyond the Kassam and Grad rockets falling on Israeli towns, attacks inside Israel like the one that killed eight students at a Jerusalem yeshiva last week, must cease first, as should the smuggling of weapons into Gaza.

But officials acknowledged that in the last few days, the rocket attacks from Gaza has tapered off significantly. Some Israeli military observers said that despite Hamas’s declaration of victory, last week’s short ground invasion, dubbed “Hot Winter” by the Israel army, inflicted much more damage on Hamas’s military installations than Israel can report publicly.

But since the assassination in Damascus last month of a veteran Hezbollah commander, Imad Mughniyeh, a killing that many in the region believe was carried out by Israel, American and Israeli officials have been closely following the calls for revenge in Damascus, Tehran, and southern Lebanon. The destroyer USS Cole and other American naval vessels that have been deployed off the Lebanese coast for the last two weeks “are there to protect our interests and to be able to deal with any contingency that might develop,” the American ambassador to the United Nations, Zalmay Khalilzad, said yesterday.