When a super PAC offers to spend tens of millions of dollars on a political candidate’s behalf, the candidate is typically grateful. That is not so in the Senate battleground of [ADDRESS], where the big-spending American Israel Public Affairs Committee has encountered some resistance, an indicator of the organization’s deepening image problems — even among Republicans — because of its staunch support for [ADDRESS]. AIPAC, a powerful pro-Israel lobbying group that is facing growing animosity in the Democratic Party from progressive critics of [ADDRESS], has spent weeks preparing to start an ad campaign to support [PERSON_NAME], the Republican nominee for Senate in [ADDRESS].

But Mr. [PERSON_NAME], and some of his allies, are wavering on the effort intended to help him, worrying that those millions of dollars could backfire politically if done so publicly. That friction, which was first reported by [PERSON_NAME] and described to The New York Times by two people briefed on the situation, who spoke on the condition of anonymity to disclose sensitive conversations, reflects the stark challenges confronting the organization as it looks to influence elections amid rising [ADDRESS] anger over [ADDRESS]’s military campaign in [ADDRESS]. Unwelcome in parts of the Democratic Party, AIPAC is now also facing concerns about its views among some Republicans, who fear that explicit public support could galvanize the lobbying group’s opponents.

Even amid the tensions, officials on both sides say they want to find a way for AIPAC to bolster Mr. [PERSON_NAME]’s campaign efforts. But precisely how remains unclear. AIPAC spent $32 million through its main super PAC, United Democracy Project, in an unsuccessful effort to defeat Dr. [PERSON_NAME], a vocal critic of [ADDRESS] and AIPAC, in [ADDRESS]’s Democratic Senate primary.

This month, which the super PAC entered with $59 million in its account, it took the extraordinary step of making plans to spend on behalf of Mr. [PERSON_NAME], who has long been supportive of [ADDRESS], and against Dr. [PERSON_NAME] in the general election. The organization, which has long considered itself bipartisan, has spent most of its money in Democratic primaries. Only once before has it spent money in a general election.

In a meeting in [ADDRESS] last week, Mr. [PERSON_NAME] made it clear to the AIPAC board chair, [PERSON_NAME], that he had significant reservations about the politics of the planned ad buy, one of the people briefed on the situation said. Mr. [PERSON_NAME] stopped short of explicitly saying he did not want [PERSON_NAME] to engage in the race, but he encouraged United Democracy Project to slow down on its media plan and give more serious thought to whether the organization is the best messenger against Dr. [PERSON_NAME], the person said. The people briefed on the situation described the impasse as not settled, and said the super PAC was still considering a pro-Rogers ad buy."The Rogers campaign has been explicit with us that they welcome support from AIPAC and our members," [PERSON_NAME], an AIPAC spokeswoman, said in a statement. [PERSON_NAME], a campaign adviser and fund-raising consultant to Mr. [PERSON_NAME], said in a statement that the Rogers campaign embraces support from AIPAC’s community."We have support from the Jewish and pro-Israel community," he said. "We want that support and will continue to welcome and are proud to have any and all support from the pro-Israel community and AIPAC community."One indication of that support: Mr. [PERSON_NAME] attended three AIPAC-hosted fund-raisers in [ADDRESS] on Sunday, and another is planned for Monday. A representative for the Rogers campaign did not respond to requests for comment. Mr. [PERSON_NAME]’s allies and AIPAC have also been considering alternate ways to support his campaign, including AIPAC members contributing to an existing pro-Rogers super PAC or funding a group of a different name so as to not disclose AIPAC as the original donor, the people briefed on the situation said.

But AIPAC leaders have privately voiced unease with a plan that makes AIP...