Saudi Arabia and Syria have asked President Reagan to help turn what has looked until now like a military defeat for the Palestine Liberation Organization (PLO) into a political victory.
Through their foreign ministers, who met here with President Reagan on July 20, the two Arab nations asked the President to go beyond the Camp David framework and give some indication that Washington endorses the idea of Palestinian self-determination. Reagan, in response, was reported to have urged Arab compromise with Israel and assistance in getting the PLO out of Beirut.
In effect, the two Arab nations were offering Reagan - and by extension, Israel - a way out of the Lebanon crisis. But their offer carried a price tag: an American pledge to move Israel toward accepting a Palestinian homeland on the occupied West Bank of the Jordan and in Gaza.
Prince Saud al-Faisal, the Saudi foreign minister, described the more than hour-long meeting which he and Abdel Halim Khaddam, the Syrian foreign minister, had with Reagan as ''friendly . . . and fruitful.'' But he declined to go into the details. Similarly, American officials declined to go much beyond generalities about the meeting, which many observers have described as crucial to a resolution of the Lebanon crisis.
What seemed clear, however, was that Reagan showed a willingness to listen to new ideas coming from the two Arab emissaries. But while he did not reject their proposals out of hand, he was clearly in a position to ask something in return. A new bargaining process now seems to have been set in motion, which could result not only in the departure of the PLO from Beirut but also in some kind of direct US negotiations with the PLO. The price for the PLO would be departure from Beirut and a clear, statement from Yasser Arafat, the PLO chairman, saying that the PLO accepts Israel's right to exist behind secure borders.
Some observers think that this delicate bargaining - being carried out under the threat of the Israeli attack on west Beirut - brings the US and the PLO closer to direct negotiations than they have been at any time since 1977 when then Secretary of State Cyrus R. Vance got the two sides close to that point.
But the Arab backers of the PLO were asking President Reagan, in effect, to give the PLO a political way out of Beirut at the very time when the organization's military fortunes seemed to be at a low point.
In the words of one Middle East specialist, this amounted to asking Reagan to make a leap of ''light years'' away from the pro-Israeli position which he has long held. Israeli Prime Minister Menachem Begin has consistently argued that a Palestinian state on the West Bank and in Gaza would constitute an unacceptable threat to Israel's security.
But the Arabs are offering to get the PLO out of Beirut without further bloodshed. They say that they will have no problem finding places for the PLO leaders and fighters to go as long as there is some hope that they can some day have their homeland. They say that the PLO is ready to go ''in order to save Beirut.'' And the PLO's exit from Lebanon would clearly reduce that organization's ability to offer a direct military threat to Israel.
One Middle East specialist here suggested that what the Arabs were looking for was a combination of a promise of real progress on the Palestinian issue from the Americans, plus some face-saving symbolism. That would make it easier to sell a PLO departure from Beirut to would-be Arab critics.
In an interview with a small group of reporters on the eve of his meeting with Reagan, Prince Saud said that he foresaw no difficulty in finding places for some 6,000 PLO fighters in the Arab world once they left Beirut, as long as this marked a ''transition'' phase before they got their own homeland. He said that Algeria and Iraq, among others, had offered to take members of the PLO in.
''There is a misconception that the Arab countries refuse to accept the Palestinians.''
But he asserted that Israel wanted to finish the Palestinian question by simply dispersing the PLO and its followers around the Middle East and then leaving them there.
''This is what is refused by the Palestinians and by the Arab countries,'' he said. ''If there is an indication that the Palestinians can go to the West Bank, everybody will accept a transition period. . . . But how can you agree to ride a train if you don't know where it's going to end?''
Saud seemed to indicate that his country is counting on changes in US public attitudes toward Israel producing a more forthcoming position on the Palestinian issue within the administration. But some experts say the Saudis and the PLO may be overestimating such changes. One expert says they may be in danger of ''overplaying their hand.''