Nearly five months after Mr. Netanyahu won a resounding victory for a third consecutive term as Israel’s premier, he faces increasing unease among Israelis as he campaigns to rally support for a congressional resolution disapproving the deal to limit Iran’s nuclear program. A vote is set for next month when Congress returns from its summer recess.JERUSALEM—The push by Israeli Prime Minister Benjamin Netanyahu to defeat the Iran nuclear deal on Capitol Hill is causing concern, even among opponents of the agreement, that the campaign could further fray the country’s ties with the U.S.

Lack of transparency by Tehran about its nuclear program and calls by Iranian officials for Israel’s annihilation have made most Israelis—including Mr. Netanyahu’s main opponents in parliament, Yitzhak Herzog and Yair Lapid deeply opposed to any agreement with Tehran.

A poll published in the newspaper Maariv after Iran and the U.S. and five other world powers signed the agreement last month indicated 71% of Israelis believe the deal brings Iran closer to a nuclear bomb. A further 47% said they supported a military strike against Iran. No sample size was given.

Yet apprehension is growing here about what is expected to be a fierce fight in Congress next month and its possible repercussions for Israel’s relations with its most important ally.

In unusually direct terms, Israeli President Reuven Rivlin this week warned Mr. Netanyahu that his aggressive campaign to defeat the deal risked harming a relationship central to Israel’s security. “The prime minister has waged a campaign against the United States as if the two sides were equal, and this is liable to hurt Israel,” Mr. Rivlin, a member of the premier’s Likud party, said in an interview published Friday in the daily Maariv. Yedioth Ahronoth and Haaretz carried similar interviews with the president.

“I have told him, and I’m telling him again, that struggles, even those that are just, can ultimately come at Israel’s expense,” said the president, adding: “We are largely isolated in the world.”

Mr. Netanyahu’s former national-security adviser, Yaakov Amidror, went on Israel Radio to defend the prime minister. The former army general said Israel’s opposition to the U.S.-brokered deal shouldn’t be subordinated to worries that it could worsen the already strained personal relationship between the Israeli leader and President Barack Obama. “There is nothing wrong with calling members of Congress,” Mr. Amidror said.

Dan Meridor, a Likud member who until 2013 served as Mr. Netanyahu’s minister of intelligence and atomic energy, urged the prime minister to look beyond the battle in Congress to long-term security interests shared by Israel and the U.S., including the containment of Iran. “We have to be able to talk with” the U.S. after the deal, Mr. Meridor said.

Liraz Mizrahi, a special-education teacher in Jerusalem, said Mr. Netanyahu should tone down what she described as his scare-mongering, stop his “false alarms and shouting,” and put the welfare of Israeli-U.S. ties ahead of the contest to defeat the nuclear accord in Congress.

In making his case for the deal, Mr. Obama has resorted to dire scenarios, too. He told American Jewish leaders at a White House meeting Tuesday in comments widely reported here that failure of the deal to win congressional approval would result in rockets being fired on Tel Aviv by Iran’s Lebanese ally Hezbollah.

For many Israelis, however, the nervousness over the looming congressional battle stems from what they see as the Netanyahu government’s deep involvement in a highly partisan, domestic U.S. political fight, said Gidi Rihat, a political-science professor at Jerusalem’s Hebrew University.

“He made the subject [of Iran] contentious and aligned himself with the Republican Party,” said Mr. Rahat, also a senior fellow at the Israel Democracy Institute. “Some of his supporters feel that he was too obstinate….They’re fatigued.”

Mr. Netanyahu’s intensive lobbying against the nuclear deal was evident again Tuesday in a Webcast hosted by the Conference of Presidents of Major Jewish Organizations, as the prime minister urged American Jews to “stand up and be counted” and oppose what he called a “dangerous deal.”

Two days later, Malcolm Hoenlein, the group’s executive vice chairman, president, told Israeli reporters in Washington that Mr. Netanyahu’s vigorous campaigning was causing “discomfort” in the American Jewish community. “Israeli governments should not be telling American Jews what to do vis-à-vis their governments,” he said. “And we shouldn’t be telling Israelis what they should do vis-à-vis their government.”

“I think that Israeli officials should make the case, on the content, on the substance, and we will do the good job of translating it into activity,” Mr. Hoenlein added.