The last thing needed by the Reagan administration was another hot war in the strategic Middle East. Now that it has one, in the Israeli invasion of Lebanon, much is riding on how and when it is resolved.

In Washington and in London, where President Reagan and his senior advisers grappled with the problem amid a European tour, U.S. statements yesterday clearly called for Israel to withdraw. The statements were less than clear on the conditions, however.

"No one knows just how to get a handle on Israel," a U.S. official in Reagan's party told Washington Post staff writer Lou Cannon, who reported a growing feeling in the traveling party that events may be out of control in the Middle East.

According to reports from Israel and statements in Washington by retired Israeli major general Nathan Sharony, who came here yesterday as a special emissary, the ambitious objective of the military operation is not only to "clean out" Palestinian fighters temporarily from a 40-kilometer belt across Southern Lebanon but also to ensure that Palestinian forces do not reenter the zone.

Sharony said he was not authorized to say how this objective could be accomplished in the absence of a permanent Israeli occupation, which he said was not intended.

He indicated, however, that Israel hopes the United States and other nations, perhaps under the aegis of an expanded United Nations force or some other international peace-keeping unit, will assume the burden from Israeli troops.

In the view of U.S. officials, such an arrangement probably would take weeks or months to negotiate, if it can be negotiated. It is likely to be made more difficult by general international disapproval of Israel's action, which is seen by many as disproportionate to the provocation.

Moderate Arab nations already are pressing the United States to use its power as Israel's patron to force a quick withdrawal.

Reagan has received messages to this effect from Saudi Arabia's King Khalid and Egyptian President Hosni Mubarak, and yesterday the ambassadors to Washington from Lebanon, Kuwait, Bahrain, Jordan and Algeria called on Deputy Secretary of State Walter J. Stoessel Jr. with much the same request.

At the same time, U.S. Jewish leaders have begun a campaign to ward off such pressure on Israel. After a meeting in New York yesterday, the Conference of Presidents of American Jewish Organizations urged the administration to reject any proposals for sanctions against Israel "for the steps Israel had to undertake in its own defense."

Secretary of State Alexander M. Haig Jr., in London, dodged the question of sanctions to bring about a withdrawal by Israel.

"There's been no decision in that area," he said. "The president has made it clear to all of us and to the American bureaucracy that he feels very strongly that our first emphasis and priority must to do all possible to bring about a termination of the bloodshed."

Whatever Haig's plans, there was no doubt that the U.S. stakes are large in the latest episode of the long-running crisis in Lebanon.

* The Israeli-Egyptian negotiations on Palestinian autonomy, a vital next step in the Camp David peace process, are likely to be set back in some degree.

At a minimum, forward movement may be delayed. The U.S. working-level negotiator in the talks, Ambassador Wat T. Cluverius, was called from the Middle East to join the presidential party in Europe after the Israeli invasion.

* U.S. relations with moderate Arab nations, already strained by nervousness about the Iran-Iraq war and charges of sluggishness and inactivity by the administration in the Middle East, are likely to become more difficult. This is especially true of Saudi Arabia and Jordan, traditionally close to the United States.

* Reconstruction and stability in Lebanon, which Haig two weeks ago described as "a focal point of danger," may be impeded, although it is too early to be certain how the Israeli action against the Palestinians will change the balance of forces in that strife-torn nation.

The cease-fire negotiated last summer by special U.S. emissary Philip C. Habib was one of the few achievements of Reagan administration diplomacy in the area, but it has been shattered and its restoration is in doubt.

* U.S. relations with Israel are at risk again, just when they were beginning to recover from the Israeli attack a year ago this week on Iraq's Osirak nuclear reactor, the Israeli bombing of downtown Beirut last summer and Israel's annexation of the Golan Heights last December. All elicited official criticism despite the administration's general disposition to support Israel.

The gravest immediate danger that policymakers foresee, and one with potential for a regionwide and even superpower conflict, is an outbreak of serious fighting between Syria and Israel. Other nations, including Syria's great-power sponsor, the Soviet Union, could be drawn in.

Such an escalating conflict is not considered probable at the moment but remains a dangerous possibility while fighting continues in Lebanon.