Richard N. Haass, president of the Council on Foreign Relations and a former senior State Department official, said it looked like Mr. Netanyahu has concluded that he would rather take his chances with the Republican Congress even at the expense of his already tense relationship with Mr. Obama.

“I really do think it represents a strategic calculation that from Israel’s point of view, this president and this White House have essentially been written off,” Mr. Haass said. “Particularly since the midterm elections, they have made the calculation that to the extent possible, they will use Congress as the channel to conduct their relationship.”

Efraim Halevy, a former head of Israel’s Mossad intelligence agency, said that calculation was made clear in December when Israel offered no public support for Mr. Obama’s decision to resume diplomatic relations with Cuba, a move he attributed to a desire not to alienate Republicans. “Israel is now placing its bets on one side of the aisle,” Mr. Halevy said. “I think it’s a mistake.”

Bernadette Meehan, a White House spokeswoman, said Friday that the relationship was bigger than any single politician or diplomat. “It is an alliance that shouldn’t be turned into a relationship between two political parties,” she said. “Our relationship is about the strong bonds between the United States and Israel and our people, and our commitment to common interests and values.”

The relationship has seen rough times under many presidents. A quarter-century ago, James A. Baker III, then secretary of state, was so angry that Israel’s deputy foreign minister had said American policy was built “on a foundation of distortion and lies” that for a while he banned him from even entering the State Department. Who was that deputy foreign minister? Benjamin Netanyahu.

By the time Mr. Netanyahu became prime minister, Bill Clinton was president and the two did not always get along either. During the 1996 campaign that first brought Mr. Netanyahu to power, Mr. Clinton made clear that he thought Shimon Peres, then Israel’s prime minister, would be the best hope for peace efforts, drawing criticism from those who saw it as interference in Israel’s elections.

It has not gone unnoticed in Jerusalem that a veteran of Mr. Obama’s campaigns, Jeremy Bird, is working with two policy groups in Israel that, while not supporting a specific candidate, are not friendly to Mr. Netanyahu. Two Republicans, Senator Ted Cruz of Texas and Representative Lee Zeldin of New York, sent a letter to Secretary of State John Kerry this week asking about any funding given to one of the organizations.