What a difference four years makes. Back in 2012, Mr. Netanyahu hosted Mitt Romney, Mr. Obama’s challenger, in Jerusalem and lavished praise on him. While Mr. Netanyahu’s team at the time denied any effort to influence the election, Mr. Obama’s camp was convinced otherwise. The rift widened when Mr. Netanyahu accepted a Republican invitation to address Congress in 2015 to assail Mr. Obama’s efforts to negotiate a deal with Iran curbing its nuclear program.

Nachman Shai, who heads a parliamentary caucus on Israeli-American relations, said Mr. Netanyahu had steered away since then from openly courting Republicans. “Because he had these tough eight years with Obama, he can’t afford it again,” Mr. Shai said. “He needs a direct line with the U.S. president.”

This spring, Mr. Netanyahu canceled a trip to Washington to attend a conference of the American Israel Public Affairs Committee, or Aipac, which aides explained by saying he wanted to avoid meeting with candidates. Last fall, amid controversy over Mr. Trump’s call to bar Muslims from entering the United States, the Republican abruptly announced and then just as abruptly canceled a visit to Jerusalem, saying of Mr. Netanyahu, “I didn’t want to put him under pressure.”

Even Mr. Trump’s Aipac speech vowing to dismantle the Iran agreement was greeted with silence in Jerusalem. “If Bibi was going in any way to support or give Trump some type of compliment, that would have been the time to do it,” Gadi Wolfsfeld, a scholar at Hebrew University, said at the time, using Mr. Netanyahu’s nickname.

Israel Hayom, the newspaper financed by the American casino magnate Sheldon Adelson, an ally of Mr. Netanyahu, championed Mr. Trump with a blast of positive coverage at the time. But like Mr. Adelson himself, who has endorsed Mr. Trump without following through on promises of large contributions, the paper has not been the unrestrained cheerleader some expected.

The left-leaning newspaper Haaretz, however, wrote this week that leaders of the Trump campaign’s efforts to recruit votes among American citizens here have ties to Mr. Netanyahu’s governing coalition. In a first for an American candidate, thee Trump campaign plans to open what it calls a “floating office” moving from home to home in West Bank settlements in coming days.

Mr. Netanyahu and Mr. Obama have tried lately to put their difficulties behind them by negotiating a 10-year American security aid package for Israel. The deal is all but complete, and the White House has been discussing how to announce it. The two leaders will both attend the United Nations General Assembly conclave this month, but it seems most likely that it would be signed by lower-level officials.