It took only two hours for President Biden, in Europe on his first overseas trip, to call Naftali Bennett after he was sworn in Sunday as the new prime minister of Israel.

A White House readout of the call and an earlier statement sent minutes after Mr. Bennett took the oath of office made one thing clear: The Biden administration is looking to start off on the right foot with the new Israel government.

Absent from the White House’s communiqués was any mention of Benjamin Netanyahu, Mr. Biden’s longtime friend whose 12-year rule came to an end Sunday. In contrast to the quick outreach to Mr. Bennett, Mr. Biden waited almost a month after taking office to call the embattled former prime minister.

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Israelis celebrated Sunday night as the parliament approved a new coalition, ending former Prime Minister Benjamin Netanyahu’s 12-year rule. His supporters gathered to cheer the country’s longest-serving leader as he takes on a new role as political opponent. Photo: Ariel Schalit/Associated Press The Wall Street Journal Interactive Edition

Mr. Biden is looking to chart a course with Israel’s new government, which is taking control at a time of intense political strife in Israel and amid growing pressure on the U.S. president from progressive Democrats to challenge Israel over the expansion of Jewish settlements in the West Bank and what they see as discriminatory policies against Arab Israelis.

Mr. Biden’s four-decade friendship with Mr. Netanyahu was seen as key to his ability to help negotiate a cease-fire in the war between Israel and Hamas last month, following 11 days of deadly fighting.

“ ‘An Israeli prime minister must be able to say no to the president of the United States on matters that endanger our existence.’ ”

— Former Israeli Prime Minister Benjamin Netanyahu

With Mr. Bennett, the Biden administration recognizes that a reboot of the U.S.-Israel alliance may be ahead, according to U.S. and Israeli officials. Mr. Bennett has never met Mr. Biden, officials said.

According to a U.S. official, the Biden administration intends to wait to see how Mr. Bennett, 49, will handle his new role as prime minister and what policies, if any, the new government might change.

Mr. Bennett is a religious right-wing politician who supports settlement expansion and is opposed to the creation of a Palestinian state. He is also opposed to America’s past involvement in the Iran nuclear deal, and on Sunday, he called talks under way in Vienna aimed at rejoining the accord a “mistake that will once again give legitimacy to one of the most violent, darkest regimes in the world.”

Born to American parents in Haifa, Mr. Bennett lived in New York for a few years to lead a technology firm, and is a fluent English speaker.

Now, he heads a brittle coalition that consists of eight parties from across Israel’s political spectrum, including a small independent Arab party that made history when it joined the government for the first time. If any of the parties pull out, the government would be at risk of collapse, and Mr. Netanyahu, who intends to stay on as opposition leader, is waiting in the wings.

The Biden administration recognizes that there is potential for Mr. Bennett to moderate politically, and some of those close to Mr. Bennett believe such a shift is inevitable, since his job as prime minister could be at risk if he alienates his more centrist coalition partners.

The U.S. official said that Israel’s new government could expect U.S. engagement “immediately,” in part, to help defuse tension in the relationship, in particular over the Biden administration’s renewed nuclear talks with Iran.

New Israel Prime Minister Naftali Bennett with a masked Benjamin Netanyahu at the Knesset in Jerusalem on Sunday.

New Israel Prime Minister Naftali Bennett with a masked Benjamin Netanyahu at the Knesset in Jerusalem on Sunday.

Photo: ronen zvulun/Reuters

Mr. Bennett on Sunday emphasized that Israel isn’t a party to the Iran nuclear agreement and “will maintain full freedom to act” in response to Iranian aggression.

Many Israeli political observers believe that to survive as prime minister, Mr. Bennett will have to maintain Mr. Netanyahu’s approach of managing the decades-old conflict with the Palestinians without trying to resolve it. It is, therefore, unlikely that he will move to annex the occupied West Bank or invade Gaza, but also unlikely he will make any major concessions to the Palestinians.

“Any Israeli leader is going to have complete jurisdiction over Palestinian lives” and Mr. Biden faces a reluctant partner in Mr. Bennett given his hard-line views on the issue of Palestinian statehood, said Noura Erakat, an assistant professor at Rutgers University and author of the book “Justice for Some: Law and the Question of Palestine.”

In remarks Sunday to the Knesset, Israel’s parliament, Mr. Bennett said his government would work to strengthen ties with both Republicans and Democrats. He also thanked Mr. Biden for Washington’s support during the recent conflict with Hamas.

In his own speech Sunday, Mr. Netanyahu said the Biden administration had asked him to keep their disagreements on Iran private, but that he refused to do so because he gave priority to Israel’s security over smooth relations with the U.S.

“An Israeli prime minister must be able to say no to the president of the United States on matters that endanger our existence,” Mr. Netanyahu said.

The White House didn’t respond to a request for comment on Mr. Netanyahu’s remarks.

Messrs. Biden and Netanyahu have a longstanding friendship that dates back to the ousted Israeli prime minister’s years as a diplomat in Washington in the 1980s. Their friendship often led to candid and sometimes heated exchanges.

There was much to disagree about: Mr. Netanyahu bitterly opposed the Obama administration’s pursuit of an international nuclear deal with Iran, and in 2015 urged a joint session of the Republican-led Congress to block the deal. Israel’s push to expand settlements in the West Bank also infuriated the Obama administration and is likely to be a flashpoint between Messrs. Biden and Bennett.

In recent years, Mr. Netanyahu’s unwavering support for former President Donald Trump —even as he questioned the legitimacy of the 2020 presidential election—has been a focal point for his critics, including many progressive Democrats in the U.S. who felt Mr. Netanyahu entrenched himself too deeply in U.S. domestic politics.

Yair Lapid, a leading centrist Israeli politician who is seen as one of the driving forces behind the new coalition and is now Israel’s foreign minister, said in an interview with The Wall Street Journal last year that one of his personal goals was to extricate Israel from America’s partisan politics.

“I thought all along the fact that Bibi (Netanyahu) put all the chips in one corner of the table, the Republican corner, was a mistake,” he said.

Write to Vivian Salama at vivian.salama@wsj.com and Felicia Schwartz at felicia.schwartz@wsj.com