“There’s a deep mutual understanding that we are what there is, there aren’t any other relationships like this, they’re all strained,” said one Israeli official who sat in on the session. “That doesn’t mean it’s going to be Bill and Yitzhak,” he added, referring to President Bill Clinton and Prime Minister Yitzhak Rabin, who was assassinated in 1995. “It doesn’t have to be Bill and Yitzhak. They get one another.”

Several people who have met with Mr. Netanyahu in recent days described him as determined and focused, the atmosphere in his office one of urgency, not panic. Since his address to the United Nations, he has hardly stopped selling his message: nine broadcast interviews in New York last week instead of the usual two or three (including radio, a first abroad since 2009, added at his request), and on Thursday, a television trifecta and rare trio of newspaper audiences targeting Britain, Germany and France.

Israeli political analysts say Mr. Netanyahu, who was educated at Harvard and the Massachusetts Institute of Technology, is action-averse but diplomatically deft, in his element behind a lectern or in front of a camera. His United Nations speech went through 50 drafts, and 45 minutes before go-time he replaced three pages near the top with a single punchy paragraph ending: “Hope charts the future. Vigilance protects it.”

Behind his desk in his office here, above a shelf filled with the encyclopedia his father edited, sit two framed photographs of men Mr. Netanyahu admires for having been able to see “danger in time” and find ways to avert it: Winston Churchill, complete with hat, pinstripes and cigar, and a long-bearded Theodor Herzl, the founder of modern Zionism.

“They were alone a lot more than I am,” Mr. Netanyahu said.

Over the past year, Mr. Netanyahu and his wife, Sara — a respected child psychologist whom many Israelis criticize for, among other things, her purported temper — have withstood mini-scandals regarding their spending on vanilla and pistachio ice cream (about $2,800 a year), makeup and hairstyling ($18,000 in 2012) and the installation of a bed for a flight to Margaret Thatcher’s funeral ($140,000). The prime minister’s stance on Iran, his signature issue, though, is popular with the public.

“Even though most Israelis dislike him, they see him as the best advocate — he knows how to deliver the goods when we are talking about talking,” said Ben Caspit, an Israeli columnist and an author of a biography of Mr. Netanyahu. “He’s a professional whistle-blower. He’s a professional prophet. But all the time pessimistic, threatening.”