Israel's charge that the United States is "taking sides" against it in the dispute over Jewish settlements in the occupied Arab territories reflects the serious breach that has developed in relations between the two allies.

Prime Minister Menachem Begin, after Sunday's angry Cabinet communique, tried to smooth over the rift yesterday, saying that "the peace process will continue" and that the dispute will not change Israel's attitude toward the upcoming efforts of Assistant Secretary of State Alfred Atherton to get negotiations moving again.

In Washington, the White House, denying that the United States is "taking sides" in the Israeli-Egyptian bargaining, said, "Our role as mediator has and will continue."

But neither side was able to resist the temptation yesterday to make its point anew on the settlement question.

Begin repeated his charge that Secretary of State Cyrus Vance had taken sides with Egypt, and said Vance's declaration that the Israeli settlements were illegal - and therefore should not exist - had "hurt us very deeply."

White House press secretary Jody Powell said the White House after much thought had "determined reluctantly that a brief comment is necessary to make sure the record is clear."

Powell then restated the U.S. position - which he said had been "repeated, publicly or privately, since September 1967" - that the United States regards the settlements as "an obstacle to peace and contrary to international law."

The seriousness of this dispute is pointed up by the fact that frequent comparisons are now being drawn to Henry Kissinger's "reassessment" of U.S. policy toward Israel in 1975, when negotiations temporarily broke down over an Israeli withdrawal in the Sinai, and to Golda Meir's angry reaction to the "Rogers Plan" for Israeli withdrawal from occupied territories put forward by secretary of state William Rogers in 1969.

The American position that Israeli civilian settlements in occupied territories are illegal has not changed in 11 years. But when Vance added that they should not exist, Israel saw a hardening of the American position at a time when negotiations with the Egyptians had reached a most delicate stage.

Some diplomatic sources saw a hardening of Israel's position in Begin's insistence yesterday that the settlements were not only legal and legitimate but "essential." They questioned whether Begin meant that Israel would not give up the settlements under any circumstances despite his claim that everything was negotiable except the destruction of Israel.

Begin is reportedly very angry at the suggestion that the credibility of his government has been compromised by its settlements policy. Yet, there are rising voices both in the United States and in Israel saying that the Begin government has not been straightforward.

Haaretz, the Israeli daily newspaper, summed up the credibility problem yesterday saying that American support for Israel since that 1967 war was based on two assertions: that the Arabs did not want peace and that Israel's main concern was security. Sadat's visit to Jerusalem "removed the ground from under the first claim," the paper said. "Now Israel's second key assertion is also being undermined. Not everyone is sure any longer that Israel is today being motivated by security concerns only . . . In Washington it is being asserted that Israeli security claims are in fact only a cover for nationalistic-biblical claims" on the occupied territory.

Begin claims, and he repeated yesterday that he told President Carter in December, that Israel's peace plan held out the "possibility that Jewish settlements in the Sinai would continue" and that there would be Israeli troops to protect them.

The Americans say Carter did not give the plan his seal of approval but said only that it contained constructive suggestions for negotiations.

According to the Israeli press, there were changes made in the plan between the time Carter saw it and its presentation to President Anwar Sadat. One such change, according to press reports, was the suggestion that Israeli settlers would remain under Israeli administration and law.

The Israelis say that Begin warned Carter that he had not yet presented his plan to his own government and that there would be some minor changes.

The Israelis say that the renewed American insistence that settlements should be removed and that there should be self-determination for the Palestinians, which Israel opposes, represents a shift toward the Arab position.

In the meantime, American Jewish leaders have been trying to tell Israeli leaders that their rank-and-file feel very uncomfortable with Israel's position on the settlements.

American Jewish leaders, according to Israeli sources, fear that their lobbying power with the American Congress is being undermined.