One by one the pieces are falling into place.
For the first time since the Lebanon crisis began, a cautious but genuine optimism over a Beirut settlement is sweeping this muggy city like a bracing breeze.
The optimism is based on apparent concessions made both by Israel and by the Palestine Liberation Organization (PLO) and on the knowledge that the chief US negotiator, Philip C. Habib, is now getting into the nitty-gritty details of a projected PLO evacuation from Beirut.
''The discussions are getting extremely detailed,'' said a US State Department official. ''That's always a good sign. . . . We're awfully close to an agreement.''
But the optimism is tempered, according to State Department officials, by a number of potential sticking points. Officials said that one of them was a continuing refusal by Egypt to take in some of the estimated 6,000 to 9,000 Palestinian fighters in Beirut unless their departure is linked to an American commitment to Palestinian self-determination. Israel's reservations about the timing of the arrival of an international peacekeeping force in Beirut are seen here as a problem that can be surmounted if other parts of the ''Habib plan'' come together.
The Israelis are against any multinational force moving in until most of the PLO is out of Beirut. This, however, is a concession in that the Israelis' former position was against such a force arriving until all of the PLO had left Beirut and Lebanon. The Habib plan calls for a part of the peacekeeping force to arrive simultaneously with the departure of the first PLO units from Beirut and provides that the force would be withdrawn if the plan weren't adhered to.
PLO officials say they want the peacekeeping force in place to protect Palestinian civilians and their Lebanese Muslim allies from reprisals by the Israelis and their Lebanese Christian allies.
The Israelis, for their part, fear that the international force could become a shield behind which the PLO might continue to stall.
The Israelis have always been slightly skeptical of official optimism emanating from Washington. Some of them believe that it can become a form of pressure on Israel -- an attempt to build momentum -- without a real justification in substance. But Israeli diplomats now agree that while the countries of destination for the PLO and the timetable of the peacekeeping force's arrival remain unresolved questions, a degree of optimism is justified.
''We would go along with a little more optimism,'' said Benjamin Abileah, an Israeli spokesman in Washington. ''But there are a lot of things that have to be cleared up.''
Mr. Abileah said that Israeli Ambassador Moshe Arens's insistence that there be a roster of names of PLO groups or individuals, showing precisely who would go to Iraq, to Syria, to Jordan, and to Egypt, amounted to a requirement that there be a full accounting for all the PLO fighters leaving Beirut. He said this did not amount to an Israeli insistence that there be a complete ''dosier'' on each of the guerrillas.
Israeli radio said Aug. 10 that the Israeli government had accepted in principle the Habib plan to withdraw the PLO.
Shimon Peres, Israel's Labor Party leader, here on a long-planned private visit to Washington on the same day, held a brief meeting with President Reagan at the White House and then told reporters: ''The President expressed his belief we are very near solving the Beirut problem peacefully.''
White House officials were adding to such comments words of caution, however. For one thing, several whiffs of optimism that arose here earlier had quickly proved ill-founded. Some American analysts are convinced, furthermore, that Israel's defense minister, Ariel Sharon, would like to sabotage the budding agreement over a PLO evacuation.
Israeli fighter planes and patrol boats this past weekend harassed an American military planning team which was working with Ambassador Habib in Beirut. The Israelis say that this was a result of miscommunication and a lack of coordination. They subsequently issued an apology to the US. But some analysts here felt the incidents -- one of which involved two Israeli fighter planes making mock attacks against two American helicopters -- did not help the process of negotiations.
One source speculated that it was simply a case of General Sharon making his presence known -- showing that he was in charge around Beirut. Another US analyst said that any further incidents like this might make it difficult for American public opinion to accept the deployment of US Marines as part of an international peacekeeping force.