President Bush, remarkably successful in guaranteeing Saudi Arabia against Iraqi invasion with his U.S.-led coalition, now courts a go-lower approach to kicking Saddam Hussein out of Kuwait, which may give ample play to the possibility of an ''Arab solution'' -- if Kuwait's former leaders agree.

Prudent caution, not the rapid decision making that marked his coalition building and dispatch of troops, is the president's likely policy now, despite the warlike trumpet he sounded in Baltimore on Monday. His personal experience in diplomacy as U.S. ambassador to the United Nations and China taught him this: putting a coalition together under a U.N. umbrella is easy compared with persuading it to drop bombs and shoot bullets.

Bush and his advisers have been unmistakably warned that members of the coalition, ranging from trusted Western European allies to the Arab leader, Egypt, want to be certain that he has explored diplomatic and political compromises with President Hussein before they give the nod to American bombs and bullets. That has not yet happened. Compromises could include such devices as a U.N.-sponsored election that Saddam might accept, despite his claim to own Kuwait.

''If the Kuwaiti ruling family bought it, we'd buy it,'' one administration strategist said privately about an election. Up to now, Bush has said Kuwait must revert to its former rulers.

Bush flexibility about how Kuwait is snatched from Saddam's hands would give Arab states a chance to find an ''Arab solution.'' Famed in the art of creative ''solutions'' to insoluble problems, Arab leaders would have maximum play in bargaining with the exiled emir for a Kuwaiti government different from the one Iraq kicked out but satisfactory to him.

Although Saddam now calls oil-rich Kuwait his own, some diplomats here believe he could change his mind overnight, just as he turned on a dime last week to make his surprising peace with Iran on Iran's terms.

Bush's performance in building his coalition and moving awesome American military power so swiftly to Saudi Arabia gets high praise from all U.S. allies, but the mood for the next step, as expressed by a key European diplomat, is far more circumspect and far less certain. ''You will need to prove that you have tried everything in the political and diplomatic field,'' he told us, before his country would sign off on any assault against Iraqi military and industrial installations.

In Cairo last week, an Egyptian policy maker conceded to us that ''the logic of the superpower will control American military actions, not our logic or even European logic.'' But, he said, the coalition of forces arrayed against Iraq had better be kept intact if the Middle East is to be put back together. That will be possible, he said, only if a military assault has wide approval.

Bush's burden in ordering force to terminate Saddam's dictatorship became much heavier when Iraq announced its perfidious hostage strategy. With up to 3,500 U.S. citizens, together with thousands of European citizens, planted at presumed targets for American bombs and shells, the president will not approve pinpoint bombing of Iraqi military installations without agonizing consideration.

The few politicians who have stayed in Washington over the congressional recess say that Saddam's gruesome hostage policy greatly complicates Bush's military option. In addition, the president is burdened by at least two other factors: How certain can the United States be that a strike against Iraq will not trigger an Iraqi counterstrike against Saudi oil installations? The answer is: not very, despite the concentration of American ground-to-air missilery now in Saudi Arabia. Officials here told us that such an attack could knock out a substantial proportion of Saudi oil capacity.

The second factor that may be pushing Bush toward caution is terrorism. No longer on the U.S. list of terrorist states, Iraq is known by American anti-terrorist specialists to be fully capable of mounting a new wave of anti-American terror whenever Saddam so decrees.

None of these factors rules out a determined and angry George Bush from deciding to shift his policy to military action, but the odds today are strongly stacked the other way. That means a lengthy period of cautious waiting while the sanctions noose tightens.