RA'ANANA, Israel—Naftali Bennett, a former chief of staff who helped Benjamin Netanyahu four years ago on his path back to the premiership, is now threatening to undermine his old boss's Mideast peace efforts.

As head of an umbrella group of Jewish settlers, Mr. Bennett says he will withdraw support for Mr. Netanyahu if the prime minister agrees to extend a partial moratorium on settlement construction in the West Bank, which expires on Sept. 26.

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