Israel’s Supreme Court faces a critical dilemma: How to handle a plan that would rein in its power.

Prime Minister Benjamin Netanyahu’s quest to rein in the judiciary has become the latest point of contention as he pushed the first part of his package through the Israeli Parliament on Monday, defying widespread protests and repeated expressions of caution from Mr. Biden.

What makes this moment different is that the rift has nothing to do with the foreign policy and national security matters that typically provoke disagreement, like arms sales, Iran’s nuclear program, territorial claims or the long-running push to forge peace between Israelis and Palestinians. Instead, it concerns a strictly domestic issue inside Israel, namely the balance of power and future of freedom in the one historical bastion of democracy in the Middle East.

The friction among friends has complicated cooperation in other areas where the two allies have common interests. For months, Mr. Biden refused to invite Mr. Netanyahu to Washington, which prevented at least some meetings between lower-level officials. The president relented last week and agreed to get together at some as-yet-unspecified time and place in the United States this year. But he then felt compelled to issue two public statements making clear that he had not changed his mind about Mr. Netanyahu’s drive to limit the power of the courts even as the prime minister is on trial for corruption.

The debate about the prime minister’s plan, which drew hundreds of thousands of protesters into the streets of Israel over the weekend in the latest of months of demonstrations, has spread to the Jewish community in the United States as well, at a time when rising partisanship has threatened to undermine American support for Israel.

“People who are left of center are worried or more upset about it overall than people who are right of center,” said Nathan J. Diament, executive director for public policy for the Orthodox Union, one of the largest Orthodox Jewish organizations in the country.

“There are many people in the American Orthodox community whose view on the substance is sympathetic or supportive to the reforms,” he added, noting that his community leans more politically conservative, “but nonetheless are worried about the divisiveness that the process has caused.”

Still, he and other longtime advocates and analysts said they remained confident that the relationship between the United States and Israel would endure. After a liberal Democratic congresswoman called Israel a “racist state,” the House overwhelmingly passed a resolution declaring the opposite was true. Only a handful of Democrats boycotted last week’s address to a joint meeting of Congress by President Isaac Herzog, and most of the rest gave him a standing ovation.

Robert B. Satloff, executive director of the Washington Institute for Near East Policy, said the fight over the judicial plan was “the clash of the century” within Israel, but it did not really affect relations with the United States in a profound way. “It’s a bit of a controversy lite,” he said. “In historical terms, this doesn’t begin to rank as a U.S.-Israel crisis.” Instead, he said, “this really is a fight within the family.”

The United States and Israel have had one of the world’s most intimate partnerships since the Jewish state was founded in 1948 and recognized minutes later by President Harry S. Truman. But conflict has been in the DNA of the relationship from the start. Every president — even the most outspoken supporters of Israel — has quarreled with Israeli prime ministers at one point or another.

Despite recognizing Israel, Mr. Truman refused to sell the new state offensive arms, as did his two successors. Dwight D. Eisenhower forced Israeli forces to withdraw from Egypt after the Suez crisis of 1956. Ronald Reagan was incensed by Israeli lobbying against his high-tech aircraft sale to Saudi Arabia. George H.W. Bush was so opposed to Israeli settlement plans that he suspended $10 billion in housing loan guarantees.

Mr. Netanyahu has been at the heart of many disputes in the last few decades. When he was deputy foreign minister, his public criticism of the United States in 1990 prompted an angry Secretary of State James A. Baker III to bar Mr. Netanyahu from the State Department. Once Mr. Netanyahu became prime minister, Bill Clinton was so turned off after their first meeting in 1996 that he asked aides afterward, “Who’s the superpower here?” using an expletive for emphasis.

Barack Obama and Mr. Netanyahu, never warm, grew even more estranged when the Israeli leader delivered an address to a joint meeting of Congress to lash out at American efforts to negotiate a nuclear agreement with Iran. Even Donald J. Trump, who bent over backward to give Israel virtually everything on its geopolitical shopping list, finally broke with Mr. Netanyahu, first over a disagreement about annexation and later over the Israeli’s congratulations to Mr. Biden for winning the 2020 election.

Mr. Biden’s relationship with Mr. Netanyahu has been scratchy going back years. Mr. Biden once said that he had given a picture to Mr. Netanyahu with an inscription using his nickname: “Bibi, I don’t agree with a damn thing you say but I love you.” As vice president, Mr. Biden was undercut during a visit to Israel by a settlement announcement. But Mr. Biden later insisted that he and Mr. Netanyahu were “still buddies.”

In some ways, Mr. Biden’s approach to Israel has been different from those of his modern predecessors. While he has reaffirmed American support for a two-state solution to the Israeli conflict with the Palestinians, Mr. Biden is the first president in decades not to pursue peace talks, a recognition that there is no short-term prospect for success.

That by itself should have been a relief to Mr. Netanyahu, who has long resented American pressure to make concessions to the Palestinians. But Mr. Netanyahu has been outspoken in his criticism of Mr. Biden’s effort to negotiate a new nuclear agreement with Iran, while Mr. Biden has called Mr. Netanyahu’s cabinet “one of the most extreme” he had ever seen.

The judicial changes have been the latest sore point. When Vice President Kamala Harris addressed a celebration of Israel’s 75th anniversary at the country’s embassy in Washington last month, just two words in her speech describing shared values — “independent judiciary” — prompted Foreign Minister Eli Cohen to snap that she had not even read the plan. Yair Lapid, the opposition leader, recently lamented that because of Mr. Netanyahu “the United States is no longer our closest ally.”

For all that, Mr. Satloff said he did not believe Mr. Biden was “looking for a fight” with the Israeli leader — leading to last week’s invitation. “My sense is the administration came to the conclusion that this tactic of withholding a presidential meeting had run its course,” he said.

Nonetheless, Mr. Biden does not think much of the judicial restructuring package, going so far as to summon Thomas L. Friedman, the New York Times columnist, to the Oval Office last week to say that Mr. Netanyahu should “seek the broadest possible consensus here.” He offered another statement to Axios on Sunday, saying that “it looks like the current judicial reform proposal is becoming more divisive, not less.”

Aides insist Mr. Biden is not trying to engineer a specific outcome in an ally’s internal politics. Jake Sullivan, the national security adviser, said the president was simply offering “judicious but straightforward” counsel.

“It’s not about us dictating or lecturing,” Mr. Sullivan said in a brief interview after an appearance last week at the Aspen Security Forum in Colorado. “It’s about us believing deeply that the bedrock of our relationship is our common democratic values.”

Other Democrats likewise said it was appropriate to weigh in with a friend. The enormous street protests “should be a cautionary note to elected leaders in Israel and I hope will give them pause,” said Senator Chris Coons, Democrat of Delaware and a close Biden ally.

But some Republicans faulted Mr. Biden for intervening in a domestic issue. “Maybe he knows more about the judicial system and he feels comfortable about telling the Israeli people what they should do,” said Senator James E. Risch of Idaho, the senior Republican on the Foreign Relations Committee. “I don’t think that’s appropriate any more than they should be telling us how we should vote on the Supreme Court here.”

In the American Jewish community, the issue has not generated the same passion seen on the streets of Tel Aviv and Jerusalem.

“The people who were very engaged in the Jewish organizational world were certainly activated by the proposed judicial reform, but I don’t think this gripped broadly the American Jewish community,” said Diana Fersko, senior rabbi at the Village Temple, a Reform synagogue in Manhattan.

Rabbi Fersko, the author of a book about antisemitism that will be released this summer, said the issue is complicated and noted deep differences between Israeli and American societies. “I don’t think the Jewish American community needs to be overly involved in this,” she said. “But I do think we need to have a deep belief that the state of Israel will find a path forward.”