Many U.S. Jewish leaders are unnerved both by the new Iran nuclear agreement and the public falling out between President Barack Obama and his Israeli counterpart, developments that are creating a rift in the durable alliance between Jews and the Democratic Party in the run-up to the 2016 elections.

Worried that Iran might still develop a nuclear weapon despite the accord announced Thursday, the Jewish leaders say they feel torn between an Obama administration that has pressed hard for a deal and an Israeli government that has repeatedly warned that Iran is a grave threat to the Jewish state and can’t be trusted to abandon its nuclear ambitions.

A group of Jewish Democratic House members met with White House Chief of Staff Denis McDonough in his office last week and cautioned that for them to help “sell a very unpopular [Iran nuclear] deal to our constituents,” Mr. Obama must “increase his popularity with our constituents,” said a Democratic congressman involved in the meeting.

Republicans have begun making moves to try to capitalize on this unease, hoping to peel away Jewish votes and campaign contributions that have historically skewed Democratic.

Republican congressional leaders have been critical of the proposed Iran deal, and the GOP’s likely 2016 presidential contenders have largely opposed it. It isn’t clear congressional Republicans have a way to block any final deal, and experts warn that even if a Republican wins the White House in 2016, it won’t be as easy to wipe off the books an international agreement as campaign rhetoric may suggest.