Mr. Ophuls, a more political filmmaker, has been well aware of questions of identity since shortly after his birth in Germany in 1927. The son of the great German-Jewish filmmaker Max Ophuls, he was forced to flee the Nazis twice as a child. The family left Germany for France after the Reichstag fire in 1933. After the German invasion of France, they made their way to Hollywood in 1941.

Unlike some of his contemporaries, that wartime experience left Mr. Ophuls wary of embracing the Zionist dream of a homeland for the Jews in the Middle East.

“I’ve come to believe that patriotism is a lie, and anyone who is a patriot is a fool,” he told Stuart Jeffries a decade ago during an interview for The Guardian conducted near his home in the French Pyrenees, close to the route his family took to escape the Gestapo. “Even though I’ve been a French citizen since 1938, most of them still think of me as a German Jew. An axe-grinding, obsessive German Jew who wants to bash France and go on and on about the treatment of Jews.”

When Mr. Ophuls failed to convince Mr. Godard that the moment had arrived this summer to begin filming, he went to Tel Aviv alone and called Mr. Sivan, suggesting that together they direct a documentary looking at both the war in Gaza and the recent rise in anti-Semitism in Europe. A central focus of the film, Mr. Sivan said in a telephone interview, is to ask if the two situations are linked. To put it bluntly, Mr. Sivan said, the two directors hoped to answer the question: “Is Israel provoking anti-Semitism?”

The film will also explore the “very strange linkage between the far right in Europe and Israel,” Mr. Sivan said. “The traditional anti-Semitic right,” he noted, “is becoming very pro-Israeli.” Indeed, many extreme nationalist politicians in Europe are strong supporters of Israeli settlements in the occupied West Bank that their governments condemn as illegal. The question the filmmakers want to explore, Mr. Sivan added, is whether “Islamophobia is the new anti-Semitism.”

The two men turned to the crowdfunding site because, Mr. Sivan said, “it’s a question of emergency,” if they are to meet a self-imposed deadline of finishing the film in time for next year’s Cannes Film Festival.

So far, the project has raised just under half of its budget, but the two men plan to return to the region next week, after a visit to Berlin to look at what Mr. Sivan, who now lives in Europe, calls the “ironic” phenomenon of young Israeli dissidents who oppose the continued occupation of the territories captured in 1967 “seeking refuge from Israel’s politics in Berlin.”