WASHINGTON — President Obama has proposed granting Israel the largest package of military aid ever provided by the United States to another nation, but he and Prime Minister Benjamin Netanyahu remain deeply at odds over a figure for the assistance despite months of negotiations.

American officials have balked as their Israeli counterparts insisted on more generous terms for a new 10-year military aid package that could top $40 billion. The divide, which could have broad national security implications for both the United States and Israel, is exacerbated by the pent-up animosity between Mr. Obama and Mr. Netanyahu, which has been stoked by their radically divergent views of the nuclear deal with Iran.

“There’s a unique place of pique for the Israelis in certain places in the administration, and I think that hovers around this negotiation,” said Robert Satloff, the executive director of the Washington Institute for Near East Policy. “It’s one of the reasons it’s taken so long to reach a decision.”

Powerful political forces are also at work. While Mr. Obama would like to burnish his legacy with an unprecedented military aid pact with Israel, some observers in the United States and Israel believe that Mr. Netanyahu is calculating that he can reach a more advantageous deal with a future president.