The U.S., pursuing a nuclear arms agreement with Iran at the time, captured communications between Mr. Netanyahu and his aides that inflamed mistrust between the two countries and planted a political minefield at home when Mr. Netanyahu later took his campaign against the deal to Capitol Hill.

The National Security Agency’s targeting of Israeli leaders and officials also swept up the contents of some of their private conversations with U.S. lawmakers and American-Jewish groups. That raised fears—an “Oh-s— moment,” one senior U.S. official said—that the executive branch would be accused of spying on Congress.

White House officials believed the intercepted information could be valuable to counter Mr. Netanyahu’s campaign. They also recognized that asking for it was politically risky. So, wary of a paper trail stemming from a request, the White House let the NSA decide what to share and what to withhold, officials said. “We didn’t say, ‘Do it,’ ” a senior U.S. official said. “We didn’t say, ‘Don’t do it.’ ”