While President Bush endorsed the continued relevance of Yasir Arafat, Prime Minister Ariel Sharon of Israel called today for exiling the Palestinian leader from the occupied territories.

Secretary of State Colin L. Powell persistently talked about hopes for a cease-fire and starting a political dialogue, even as Israel broadened its military campaign to rout terrorism suspects in Palestinian areas and indicated that the current operation would last weeks.

But as the suicide bombings show no sign of abating and the Israeli military response intensifies, a growing chorus of experts and critics across the political spectrum have put forth a range of policy options they say could advance the chances for a peace settlement.

So far, the administration is holding to longstanding proposals to first cement a cease-fire and then begin talks that could lead to a negotiated settlement. The issue is, should it continue to do so as violence rages?

The United States has three basic choices, experts say. One is to give the Israelis a green light to evict Mr. Arafat and dismantle the Palestinian Authority. This approach is based on the assumption that Mr. Arafat will never be a reliable negotiating partner and that Israel has no choice but to act alone to constrain terrorism.

A second approach — essentially the administration's current posture — is to watch the Israelis' military offensive from the sidelines, while cautioning them not to harm or evict Mr. Arafat. The hope is that the two sides, exhausted and bloodied, will eventually sit down with the administration's special envoy, General Anthony C. Zinni, and begin talks on establishing a truce under guidelines devised last year by George J. Tenet, the director of central intelligence.

The two sides would then proceed to talks on political questions, guided by the recommendations of an international committee headed by former Senator George J. Mitchell, which would aim at freezing Israeli settlements and taking other steps to ease the confrontation. They would also open talks on a final settlement that would create a Palestinian state to coexist peacefully with Israel.

A third approach — one increasingly being proposed with different variations in Washington — is to insist that political issues be dealt with at the same time as the security ones. This approach is based on the assumption that the Palestinians will not give up terrorism until they see they can make real gains toward a Palestinian state, while the Israelis will not begin political talks until the terrorism ceases.

"Something more decisive from the United States is probably necessary," said Dennis B. Ross, the former negotiator for the first Bush administration and the Clinton administration. "You can't continue to dribble out your package."

Mr. Ross has proposed that the United States announce the steps both sides would need to take to stop the fighting, which would include an end to Palestinian terrorism as well as Israeli troops withdrawals, and when the actions should be taken. The United States would also specify a schedule for beginning political negotiations on a final settlement and outline the agenda. Foreign ministers from the Middle East would be invited to Washington to give the effort a push.

By putting more emphasis on political talks, the United States would pose a truer test of Mr. Arafat's intentions, Mr. Ross said. But he cautioned it might yet turn out that Mr. Arafat is not willing or able to negotiate in good faith.

Zbigniew Brzezinski, Jimmy Carter's national security adviser at the time of the first Camp David accords, would go further by drafting a potential Middle East settlement in Washington, pressing both sides to accept it and enforcing it with American peacekeeping troops. Mr. Brzezinski said that the Bush administration has essentially "winked" at the Israeli offensive and asserted that it was likely to make it more difficult to resume negotiations.

"I fail to see how plunging the Palestinian side of the equation into anarchy is a contribution to the peace process," Mr. Brzezinski said. "That is what our policy amounts to."

So far, there seems to be little appetite at the highest levels of the administration for Mr. Ross's approach and no interest at all in Mr. Brzesinski's call to impose an settlement, although the State Department's Middle East experts have long pressed for a more political element to negotiations on security.

"I don't see that getting any traction over at the White House," one administration official said today, referring to Mr. Ross's approach. He noted that granting the Palestinians' demand for such simultaneous negotiations could be seen in some quarters as rewarding suicide bombers.

At the same time, however, officials say there is equally little push for cutting off relations with Mr. Arafat, though Vice President Dick Cheney and Defense Secretary Donald H. Rumsfeld are seen as having little patience for his tactics and have less hope that he can be a trusted negotiator, in light of Mr. Arafat's failure to meet Mr. Cheney's conditions for a meeting with the vice-president and the Palestinians' attempts to buy arms from Iran.

Officials say there are too many uncertainties about who would succeed Mr. Arafat, among other questions.

So as the fighting on the ground grows toward an uncertain end, the administration has straddled an uneasy middle ground between unleashing the Israelis — allowing them to destroy the Palestinian Authority once and for all — and rushing to stop the violence and promote a political settlement.

"What they're doing is routing out terrorists," Secretary said today, referring to the Israeli offensive, on the ABC program "Good Morning America." And we understand that. But we also understand that, as they say, they will leave these areas in due course, and at that point, we've got to have a process ready to move forward, to get into a cease-fire, and into political discussions, and that's what we're focusing on."

The Bush administration's posture has not been without some support. Backing some aspects of the administration's policy, former Secretary of State Henry A. Kissinger has argued that the administration should not aim for a final settlement right away. If the parties will not commit themselves to such a course, he says, the United States "has no option except to stand aside."

Still, the administration's wait and see policy seems to be contradictory in some respects. The administration, for example, insists it is not giving Israel a green light to send its army into Palestinian cities but at the insistence of Washington the United Nations resolution adopted last weekend does not specify a time table for withdrawal.

Bush administration officials insist over and over that Mr. Arafat is not doing enough to rein in terrorist attacks, but they have not warned they will cut off cut relations if he fails to act.

Mr. Cheney said last week that the two sides were not able to achieve a peace by themselves and that an American role was essential.