I'll offer up another completely unsubstantiated theory, simply Israeli intelligence (Mossad) trying to swing dick on Dulles' CIA. Basically "you're not untouchable in our neck of the woods."
Interestingly enough, 14 days before this...:
After a tense confrontation with CIA Tel Aviv
station chief John Hadden on May 25, 1967, who warned that the United States would help defend Egypt if Israel launched a surprise attack, Mossad director Meir Amit flew to Washington, D.C. to meet with U.S. Defense Secretary Robert McNamara and reported back to the Israeli cabinet that the United States had given Israel "a flickering green light" to attack.[33]