When Israeli Prime Minister Shimon Peres made an unofficial visit to Washington the other day, he praised the Reagan administration for its move against Libya's Colonel Qaddafi. Secretary of State Shultz praised Peres for Israel's handling of its economic problems. On the face of it, you had to wonder whether a phone call wouldn't have been enough.

But as is often the case in the Middle East, there was more than met the eye -- including a wind-up, one-on-one breakfast conversation between Peres and Shultz, which went well beyond the publicized, official encounters. From both Israeli and American officials, I have the impression that Peres and Shultz took the occasion to explore in some depth -- I wouldn't make it more decisive than that -- some new and different ways to try to break the deadlock over the Arab-Israeli conflict by attempting to treat it in some wider context.

The nub of it remains the fate of the Israeli-occupied West Bank and Gaza. But the current brainstorming in Washington and Jerusalem includes the problem of Palestinians in refugee camps in Lebanon; for example, it includes, as well, the politically destabilizing economic distress in key Arab countries (notably Egypt and Jordan) as a consequence of plummeting oil prices.

Most of this probing is highly tentative, which might lead you to ask: what's new? The answer is that Shimon and the Reagan administration share with another key player, Jordan's King Hussein, an acute awareness of the same deadline. In October, the Labor party gives way to its coalition partner, the Likud; Peres switches jobs with Yitzhak Shamir, now foreign minister, for the last two years of the four-year government of "National Unity."

But Shamir's power derives from his leadership of the Herut party, which is caught up in a bitter power struggle. Peres is riding high in public- opinion polls. With the right issue, he could conceivably crack open the "National Unity" coalition and bring about an election that could give his Labor party a working majority -- before October.

Clearly, both the Reagan administration and King Hussein would rather do business with Peres, who has been remarkably forthcoming in his efforts to revitalize a "peace process." Shamir, who voted against the Camp David accords, has vigorously resisted almost every conciliatory gesture by Peres.

The Reagan administration's stake in an Arab-Israeli-Palestinian settlement rests in part on its own investment in the so-called Reagan Initiative of September 1982. It rests as well on a sensible realization that unresolved conflict in the Middle East is an invitation to instability, political upheavals and violence, if not outright resumption of hostilities.

Hussein has twice failed to bring the PLO's Yasser Arafat into acceptance of preconditions for a PLO negotiating role, which would satisfy even U.S. requirements, let alone Israel's. Yet, for Hussein to go it alone would put him at even greater risk at the hands of Arab extremists. Not too many weeks ago, the King was thought to be souring on any hope for negotiations and sulking over the Reagan administration's withdrawal of its proposed arms sales to Jordan when they faced certain defeat in Congress. But U.S. officials say there are some signs that Hussein has not yet given up, the more so if there is any hope for resolving the question of Palestinian representation in any renewed peace talks before a changeover of Israeli leadership slams the door.

Shultz is said to have developed a deep appreciation for Hussein's dilemma, as well as an admiration for his efforts to resolve it. But there is no reason to believe that Arafat or some alternative Palestinian representation can be successfully brought into new peace talks.

Obviously, somebody will have to make a move. One possibility: a unilateral Israeli initiative, including an offer to draw down the Israeli military presence on the West Bank and a larger grant of "home rule" in matters having to do with health and welfare, education and water resources. Peres' coalition partners would violently object. But that could provide him with the "obstructionist" issue that would break up "national unity" and bring on elections.

Such is the Israeli concern over the potential for political upheaval -- as extremists exploit economic distress -- that Peres has been sounding out European leaders as well as the Reagan administration on a long-term international economic development program for the region. The idea would be to call on the Europeans and Japanese, who are profiting the most from low oil prices now. They would have the most to lose at the hands of rampant Arab radicalism over the long haul when their dependence on Mideast oil will be largely undiminished but the price may have gone back up.

That none of this is easy to get a handle on takes nothing away from an encouraging sense that some of the key Middle East players are responding to the hard rule of Middle East life: when nobody is doing anything positive, things tend to disintegrate in unpleasant, negative ways.