For Egypt, the stakes are high.
If its diplomacy succeeds on behalf of the Palestinian Liberation Organization (PLO), it will help Egypt reestablish links with a number of key Arab nations.
If its diplomacy fails, the result could be an intensification of popular unhappiness with Egypt's President Mubarak. Because of Egypt's economic difficulties, the honeymoon which Mubarak enjoyed when he first took power is now over.
Reports from Cairo indicate that many Egyptians have grown disillusioned with the Egyptian-Israeli peace treaty of 1979 and are sharply critical of the Israeli invasion of Lebanon. Egypt is apparently the only Arab country to have seen sizable demonstrations protesting the invasion.
If Mubarak were to look in any way like an accomplice of the Israelis, the demonstrators could turn against him. The Egyptians, therefore, have a strong incentive to try to help prevent an Israeli assault on Beirut and to help obtain political concessions for the men of the PLO before they leave Beirut.
So far, Egypt has not obtained the concessions it seeks from the United States - either US agreement to a direct ''dialogue'' with the PLO, or a formal US endorsement of the Palestinians' right to ''self determination.''
In the diplomatic language used by Middle East specialists, self-determination is generally interpreted to mean an independent Palestinian state on the Israeli-occupied West Bank of the Jordan River and in the Gaza district. It is a concept which is strongly opposed by the Israeli government of Prime Minister Menachem Begin.
However, Egypt is not only the most heavily populated of the Arab nations, but also the strongest of the Arab countries which are friendly to the United States. Its views cannot be ignored. And it has increased its bargaining power by offering, conditionally, to take in some of the PLO leaders and fighters now encircled in West Beirut.
In an interview with the Monitor following his talks here at the end of last week with President Reagan and Secretary of State George P. Shultz, Egyptian Foreign Minister Kamal Hassan Ali said that Egypt has agreed ''in principle'' to accept into Egypt an undetermined number of PLO fighters if ''some hope'' is given to the PLO for its political future. Syria, Jordan, and Iraq have also apparently given their conditional agreement to taking in some of the estimated 6,000 to 9,000 PLO fighters in Beirut.
But Mr. Ali said that if the PLO fighters are simply transferred to four Arab countries and given no hope for the future, they will be ''radicalized'' and constitute a new danger to stability in the region. As it is, the Egyptian foreign minister contended that 80 percent of the PLO consists of ''moderates.'' He described PLO chairman Yasser Arafat as the leader of the moderates.