Israel's immediate and strong rejection of President Reagan's proposals for addressing the Palestinian issue in the Middle East is likely to inaugurate a time of troubles in Israeli-American relations, administration sources said yesterday.

On the other hand, the Reagan initiative is likely to provide an opportunity to repair recent and severe strains in U.S. relations with Egypt, Jordan, Saudi Arabia and some other Arab states, the sources said.

Judging from the initial reaction, however, neither the deterioration of U.S. relations with Israel nor the improved ties with the Arabs will be automatic or immutable. Officials believe there is considerable room for maneuver in the positions of both sides in response to the Reagan plan, depending in large part on unpredictable political and military developments in the region.

The sharp reaction of the government of Prime Minister Menachem Begin was expected here, in view of past Israeli stands on Jewish settlements on the West Bank and other issues mentioned by Reagan. Washington officials were relieved that Israel reacted by talking rather than acting, considering some predictions that Begin would announce new settlements, annex the West Bank, or pull out of the peace negotiation process.

Moreover, some reports from the U.S. Embassy in Tel Aviv suggested that not all the Israeli establishment is completely unhappy. Some relatively positive soundings, as expected, came from the opposition Labor Party, but there are also stirrings reported within the Begin camp.

Unquestionably, Reagan's declaration ruling out an independent Palestinian state was pleasing to Israel. In the view of some influential Israelis, as reported to American diplomats, most of the rest of Reagan's views had been advanced by the Ford or Carter administrations or both, and Israel had found ways to live with them. There is no doubt, however, that for the foreseeable future U.S. relations with Israel have been complicated by the surprise announcement of Reagan's proposals. After appearing to handle Israel with kid gloves during its first year and a half in office -- the tenure of Alexander M. Haig Jr. as secretary of state -- the Reagan administration is taking a much more independent position with George P. Shultz at the foreign policy helm.

One official forecast "a long-term period of sustained difficulties" between Washington and Jerusalem. Israeli pride as well as Israeli policy has been dealt a blow, right after a triumph of arms in Lebanon that vanquished the Palestine Liberation Organization and changed the flow of power in Lebanon and the Middle East.

Based on the positions of Haig and some others in the U.S. government, the Israelis had reason to expect a vote of confidence and thanks from Washington rather than a new, independent and troublesome tack.

The Arab reaction to the Reagan plan is the other side of the coin. As usual, it was slower to develop and much less clear-cut than that in Israel, but the initial soundings were read here as promising.

In the view of key Arab leaders, confidence in and relations with the United States sunk to the lowest point in years during the Lebanon fighting, which was widely seen among Palestinians and Arab peoples as the result of U.S. collusion with Israel.

Leaders in Saudi Arabia, especially, were very agitated. During the early weeks of the fighting in Lebanon, King Fahd sent word to Washington that the U.S. position in his country and throughout the Arab world was hanging in the balance. Other senior Saudis sent word that a close relationship with the United States was becoming a distinct liability rather than an asset, and some of them were reported to be afraid for the first time of assassination or other violence against those seen to be friends of Washington.

Shortly after Haig's resignation late in June, according to authoritative sources, Fahd told an American visitor in a message intended for Washington that the close commercial and financial relations with the United States, involving 500 U.S. companies doing business there as well as large Saudi investments in the United States, "won't last long" if the United States continued its existing course in Lebanon. The Saudi king did not explicitly threaten to break the U.S. connection but his views implicitly suggested such a possibility, the sources said.

President Hosni Mubarak of Egypt and King Hussein of Jordan also felt threatened to an unprecedented degree by the widespread belief in U.S. complicity in the Israeli military action and, in the case of Mubarak, by his vulnerability to the charge of Egyptian complicity. With this background, U.S. diplomats in the area regarded Reagan's backing for "Palestinian rights" and especially the independence of his position from that of Israel as essential to restoration of an American position with moderate Arabs.

Shultz said yesterday that Hussein is taking the Reagan proposals seriously and is studying them, but that "he has made no commitment" to accept Reagan's plea to become actively involved in the peace process. Other officials said there is little prospect that Hussein would act on his own without backing from other key Arabs.

In this respect the Arab summit expected to convene in Fez, Morocco, next week will be closely watched for signs of a developing consensus about the Reagan plan and the peace process. The reaction of the PLO is likely to be particularly important. The other Arab leaders, who were unwilling or unable to rescue Yasser Arafat and his men in Lebanon, are expected to make up for their inaction, at least rhetorically and symbolically, by rallying around the PLO leader at Fez.

Some skeptical Arabs are reported to be already asking U.S. diplomats what Washington will do in the face of Israeli opposition. These Arabs have become convinced on the basis of recent history that American administrations back down in the face of Israeli objections and, in other cases, take no counteraction when Israel acts in its own interest in ways displeasing to Washington.

In the political chemistry of the Middle East, the negative Israeli reaction to the Reagan plan seems likely to precipitate a more positive Arab reaction than otherwise might come to the surface. But if the Israelis should mute their opposition and begin to see some advantages in the Reagan plan, the Arabs are likely to become wary or even negative, in the view of administration experts.