UNITED NATIONS — Officials in Lebanon, Israel, Washington, and at the United Nations who were expecting to turn to Paris to make up the “backbone” of a U.N. force that is expected to be deployed in Lebanon have shifted their sights to Rome.

But even as Prime Minister Olmert said Italy should “lead the international force,” and as Prime Minister Siniora of Lebanon also welcomed Italy, the definition of the force’s mission was not clear enough to persuade many in Europe to agree to send troops to Lebanon.

President Bush said yesterday that the Security Council might need to pass a new resolution to make the need to disarm Hezbollah clearer. “There will be another resolution coming out of the United Nations, giving further instructions to the international force,” he told reporters at the White House.

“We’ve always contemplated the disarming of Hezbollah, which was not specifically addressed in 1701, would have to be addressed, and that should be coming shortly,” the American ambassador to the United Nations, John Bolton, added, referring to the council resolution that led to the lull in Israel’s war against Hezbollah.

But after the long, bruising diplomatic fight over that resolution, no one seemed in a rush to begin a new round of discussions over a new resolution. “I don’t think we have a particular view on time,” Mr. Bolton said, and added, “There’s no reason that it should hold up the deployment” of troops.

Prime Minister Prodi of Italy called Secretary-General Annan yesterday, and, separately, Mr. Olmert. According to Italian press reports, Mr. Prodi offered between 2,500 and 3,000 troops for the new force, including some to be deployed in the first wave.

It is “important that the force arrive as soon as possible,” Mr. Olmert said, according the prime minister’s Web site. Italy, he added, should “lead the international force, as well as send troops to oversee the border crossings between Syria and Lebanon.”

So far, the U.N. peacekeeping department has concentrated on trying to assemble a force of 3,500 troops to be deployed by September 2 exclusively in southern Lebanon. But Jerusalem insists that to stop the rearming of Hezbollah, the international force should also deploy on the Syrian border.

Lebanon’s defense minister, Elias Murr, said yesterday that his army would prevent Hezbollah from launching any further cross border attacks. “Any rocket that is fired from Lebanese territory would be considered collaboration with Israel,” he said. Israel, however, called on the international force to back up the weak Lebanese army.

U.N. planners were stunned when, after statements made by leaders in Damascus and Tehran, the French Defense Ministry last week announced it would contribute no more than 200 engineers in the first phase.

To save face, France convened a European Union meeting, scheduled for tomorrow in Brussels, to discuss the Continent’s contribution to the new force, which is hoped will reach its full capacity of 15,000 troops in 90 days.

Unlike the timid participation offered by Europe so far, Muslim countries that maintain no relations with Israel, such as Indonesia, Malaysia, and Bangladesh, have offered troops. However, Mr. Olmert said countries that maintain no diplomatic relations with Israel should not participate in the force.

“It’s a long-standing custom in U.N. peacekeeping that the parties have to agree on who the troop contributors will be,” Mr. Bolton said, calling Mr. Olmert’s position “pretty straightforward.” Countries that “don’t even recognize the state of Israel can’t be counted upon to fulfill the impartial role that we look for in the enhanced UNIFIL,” he said.

U.N. officials, however, maintain that Mr. Annan is responsible for composing the new force, and that since it will be deployed on Lebanese soil, Israel should have no say in the participants.