ASHINGTON, Sept. 18 ・After a week of unconditional support from abroad, the Bush administration confronted its first significant difficulties today in building a broad international coalition to support using military power and other means against a still-faceless terror network rooted in Afghanistan and elsewhere.

A procession of world leaders was either on the way or on the phone to Washington seeking to convince the White House that only a multilateral approach based on consultation, hard evidence and United Nations support would justify the use of military power in response to the devastating attacks last week.

Today, President Jiang Zemin of China telephoned Prime Minister Tony Blair of Britain and President Jacques Chirac of France as each prepared for meetings with President Bush. He admonished his Western counterparts to tell Mr. Bush that "any military action against terrorism" should be based on "irrefutable evidence and should aim at clear targets so as to avoid casualties to innocent people," according to official news reports from China.

Mr. Jiang also telephoned President Vladimir V. Putin of Russia, and although the two leaders denounced "terrorism in all its forms," they spoke just of cooperating with each other and the United Nations to "develop a mechanism for fighting terrorism," the reports said.

As the Bush administration sought through White House consultations and overseas missions to strengthen the sinews of an antiterror effort whose scale and objective remain unknown, a number of countries began to calculate the potential cost of their taking part, and to try to exact a price for it from the United States.

For a number of Middle Eastern countries, the price was straightforward. The United States has to become more deeply involved in ending the violence and in reinvigorating the Israeli-Palestinian peace effort.

But it was clear that a convulsion in Israel, the West Bank or Gaza could threaten Washington's efforts to maintain support in moderate Arab countries, a problem that Mr. Bush's father faced in the 1991 coalition that defeated Iraq in Kuwait.

"The people that we expect to work with closely in combating terrorism," a spokesman for the State Department, Richard A. Boucher, said, are "interested in the Israel- Palestinian situation," and their attitudes toward America's war on terrorism are "linked in people's minds" to America's commitment to Arab-Israeli peace.

Foreign Minister Saud al-Faisal of Saudi Arabia, is due to arrive on Wednesday with a large contingent of Saudi intelligence officers and their files on Osama bin Laden and his Al Qaeda network.

But other potential American allies raised urgent economic and political agendas that officials said Washington was beginning to address. Pakistan, in exchange for whatever bases or rights to fly in its air space that it provides, would like an agreement to end 11 years of sanctions, to restore the flow of American arms and to reduce a punishing debt load.

Russia, if it is called on, has a clear set of grievances over NATO expansion toward its borders and criticism of its military campaign in Chechnya. Foreign Minister Igor D. Ivanov arrives on Wednesday. Administration officials said they were eager to establish Moscow's price to open the northern corridor to Afghanistan through Tajikistan, a former Soviet republic.

A number of Russian generals have questioned whether Russia could join an American-led antiterror campaign whose operational objectives remain unclear. One high- ranking military officer told a newspaper, Vremya Novestei, that "fighting terrorists is like trying to rid oneself of roaches in a block of flats."

"You do it in one flat," the officer said, "and they go to another."

Nowhere was the sense of alarm over American plans more apparent than in the warning of one of America's staunchest Middle East allies, President Hosni Mubarak of Egypt. In remarks broadcast on Monday night, he implored the United States not to undertake military action that might kill innocent civilians, divide Christians against Muslims and further inflame attitudes against American policy in the region.

Mr. Mubarak, like Mr. Jiang, urged that "hard evidence" be the basis for any military action and that "countries not be punished" for the actions of "individuals." He called on the United Nations to organize an international convention against terrorism that would develop a common program of action for all countries.

His remarks were echoed by other leaders in the region where Washington has yet to establish a firm diplomatic beachhead in dealing with intractable and volatile conflicts.

While Egypt and Jordan were both crucial allies in the 1991 coalition against President Saddam Hussein of Iraq, diplomats from both countries said they did not expect to be called on to provide bases or other direct military support. Both said they were providing intelligence information on terrorist groups to the Central Intelligence Agency under longstanding agreements.

Beneath the veneer of solidarity and support in Europe, misgivings can be heard about how Mr. Bush plans to proceed. Germany has repeatedly called for a multilateral approach to the problem and warned against America's going it alone.

Speaking at the White House today, Mr. Chirac pointedly declined to accept Mr. Bush's characterization of the fight against terrorism as a war. "I don't know whether we should use the word `war,' " the French leader said.

Diplomats noted that Mr. Bush sent a high-level State Department envoy, John R. Bolton, to Moscow on Monday to push forward on American missile defense plans, even though a decision by Mr. Bush to withdraw unilaterally from the Antiballistic Missile Treaty of 1972 would raise questions of a return to a "go it alone" ethos in international affairs.

President Bush's father last week seemed to be the first to declare dead the sort of unilateralism that prevailed in the administration's early months. He told a Boston audience, "Just as Pearl Harbor awakened this country from the notion that we could somehow avoid the call to duty and defend freedom in Europe and Asia in World War II, so, too, should this most recent surprise attack erase the concept in some quarters that America can somehow go it alone in the fight against terrorism or in anything else for that matter."

No one has suggested, least of all the former president, that his statement represented criticism of his son or the current administration. But it seemed an unmistakable effort by the father to assert that the son was breaking with the recent past.

If policy is changing, nobody seems quite sure where it is heading. Just what Mr. Bush, Secretary of State Colin L. Powell and Vice President Dick Cheney meant when they indicated that harboring terrorists would be a casus belli in the fight against terrorism remained unclear.

In Moscow, an influential parliamentarian, Aleksei G. Arbatov, said although the consensus there was "total moral support" for the United States and the struggle against terrorism, there also existed a strong humanitarian concern "not to resort to massive strikes, to nonselective actions which are unjustified from the moral point of view, to avenge the death of thousands of innocent people with the deaths of tens of thousands of other innocent people."

Karl Kaiser, a foreign policy expert in Germany, said the "experience of the first months of the administration caused a great deal of concern in Europe about unilateralism."

"However," Mr. Kaiser said, "something rather extraordinary has happened, and the reaction of the administration thus far, contrary to some fears that existed, was so different, so cautious and stressing the need to act with others." As a result, Mr. Kaiser suggested that at least for now "the image of the cowboy shooting from the hip is gone."