A secretive U.S.-backed initiative to forge closer ties between Saudi Arabia and Israel faces setbacks after the crown prince, who spearheaded the effort, was implicated in a journalist’s killing along with two of his aides.

The two close advisers to Crown Prince Mohammed bin Salman, who played key roles in the behind-the-scenes contacts between the two countries, lost their jobs over suspected involvement in the operation that led to journalist Jamal Khashoggi’s death inside the kingdom’s Istanbul consulate in October....