Speaker Kevin McCarthy offered on Monday to host Prime Minister Benjamin Netanyahu for high-level bipartisan meetings in Congress — issuing an implicit challenge to President Biden, who has refrained from welcoming the Israeli leader to the White House in a protest against his domestic agenda.

The offer fell short of a formal invitation, but the comments were a break with diplomatic custom and tradition, and reminiscent of a similar move by congressional Republicans during the Obama administration when tensions in the U.S.-Israeli alliance were similarly fraught.

The move also risked exacerbating tensions between Democrats and Republicans over how the United States should manage its alliance with Israel, a bond that has traditionally had bipartisan support in Washington.

Mr. McCarthy’s comments, at a news briefing after he spoke at the Israeli Parliament, suggested a business-as-usual approach to Mr. Netanyahu’s government.

Mr. Biden, a Democrat, has taken a different tack, declining to invite Mr. Netanyahu to the White House as a signal of displeasure at the Netanyahu government’s now-suspended plan to overhaul the Israeli judiciary, which has set off widespread protests in Israel as well as prompting unease within the Jewish diaspora.

The judicial overhaul includes proposals to give the government more power over the selection of Supreme Court judges and to allow Parliament to override the court’s rulings. Critics of the plan said it would significantly undermine the strength of Israel’s democracy; Mr. Biden had criticized the government’s previous efforts to move ahead with the plan without seeking society-wide consensus.

Asked if he would host Mr. Netanyahu if Mr. Biden did not, Mr. McCarthy said: “Yes.”

“I have a long relationship with the prime minister, the longest serving prime minister of Israel,” Mr. McCarthy said.

Later he said at the briefing: “The prime minister should come and meet with members” of Congress.

His comments followed an interview with Israel Hayom, a right-wing newspaper published by the Adelson family, donors to conservative causes in both Israel and the United States, in which Mr. McCarthy said that if an invitation from Mr. Biden “doesn’t happen, I’ll invite the prime minister to come meet with the House. He’s a dear friend.” Mr. McCarthy confirmed at the briefing that those comments had been quoted accurately.

Mr. McCarthy was visiting Israel at the head of a bipartisan delegation of 17 members of Congress, including Steny H. Hoyer, one of six Democrats on the trip. In a nod to bipartisanship, Mr. McCarthy also said he would invite Hakeem Jeffries, the House minority leader, to meet with Mr. Netanyahu during any visit by the Israeli prime minister to Congress.

Nevertheless, such a visit would risk be exceedingly awkward for the White House given Mr. Biden’s resistance to meeting with Mr. Netanyahu. The situation could draw comparisons with a similar one in 2015, when Mr. Netanyahu accepted a formal invitation to deliver a joint address to the Republican-led Congress, circumventing the Democratic-held White House.

Republican leaders extended the invitation to Mr. Netanyahu during the height of his protest against the Obama administration’s efforts to strike a multilateral international agreement to constrain Iran’s nuclear ambitions, efforts the Republicans also opposed.

A rare breach of protocol, that move was seen as a snub to President Barack Obama, and was described as an “insult” by Representative Nancy Pelosi, the Democratic leader at the time. Scores of Democrats avoided the event.

U.S. support for Israel is considered essential to its security. Washington provides more than $3 billion in military aid every year, as well as providing diplomatic support for Israel at the United Nations and helping to build key military infrastructure, like the Iron Dome air defense system that protects Israelis from rocket fire from armed groups in Gaza.

Bipartisan enthusiasm for that assistance is considered essential to sustaining such a high level of support.

Mr. Netanyahu has long indicated his preference for the Republican Party, appearing in 2012 to support Mitt Romney, that year’s Republican challenger to Mr. Obama. He also formed an unusually strong relationship, until recently at least, with Mr. Obama’s Republican successor, Donald J. Trump, further threatening bipartisan support in Washington for Israel.

But hopes were raised this year of a warmer relationship with the current president; though Mr. Biden served as Mr. Obama’s vice president, he often boasts of his decades-old friendship with Mr. Netanyahu, which he managed to sustain throughout the Obama presidency.

That rapport has been frequently tested this year. In addition to criticizing Mr. Netanyahu’s judicial overhaul proposal, the Biden administration has criticized other comments or actions by members of his far-right government — the most nationalist and religious in Israel’s history.

The government’s stances have also strained ties between Israel and the Democratic Party at large, as well as with liberal American Jews.

In his remarks, by contrast, Mr. McCarthy referred only in passing to Mr. Netanyahu’s judicial plan and the need for checks and balances in a democracy. In an implicit criticism of Mr. Biden, he also said it was up to Israelis to decide what form their judiciary should take.

“Everybody would agree there’s some form of reform that they believe they need to have, but we leave it up to you, to your country, to figure out how to do that,” Mr. McCarthy said.

“Israel is their nation, Israel can decide what they want to do,” he added.

The Republican Party has long tried to position itself as the more pro-Israel party in American politics, partly in an attempt to win a bigger share of Jewish and Evangelical voters. That includes fostering a close relationship with Mr. Netanyahu, even as his leadership has grown more contentious.

Last year, Mr. Netanyahu addressed a gathering of the Republican Jewish Coalition just weeks before returning to power in Israel; in recent weeks, leading Republicans defended Mr. Netanyahu against criticism from Mr. Biden over the proposed judicial changes in Israel, chastising the president for opining on another country’s domestic debates.

But even as Mr. McCarthy spoke out on Monday, strong signs of bipartisan support remained. He made his comments at the briefing while standing beside Mr. Hoyer, who had moments earlier praised the speaker for a speech he made to the Israeli Parliament that afternoon. Their visit followed a similar tour late last month by the House minority leader. Mr. Jeffries, Democrat of New York, spoke during that visit of “our strong support for a Jewish and democratic state.”

Mr. McCarthy and Mr. Hoyer also visited Israel together in 2019, pledging during that trip to keep domestic U.S. politics out of the U.S.-Israel relationship.

Last week, Mr. Netanyahu appeared aware of the dangers of annoying both Democrats and Republicans when he agreed to meet with Gov. Ron DeSantis of Florida, a rival of Mr. Trump and a potential challenger to Mr. Biden — but avoided releasing a statement about their meeting.

Over 90 percent of House Democrats also voted to back a resolution last week celebrating the 75th anniversary of Israel’s founding and calling for an expansion of the Abraham Accords, the set of agreements brokered by Mr. Trump in which three Arab states moved to normalize relations with Israel, a copy of which Mr. McCarthy proudly wielded during his speech at the Parliament.

Yet Democrats have expressed wariness about some of those developments. Last week, a group of leading House Democrats, including several of its most prominent Jewish members, released a statement expressing chagrin that the document, “principally drafted by Republicans, broke the longstanding bipartisan tradition of acknowledging the importance of achieving a two-state solution between Israelis and Palestinians.”

Myra Noveck contributed reporting from Jerusalem.