WASHINGTON—U.S. intelligence agencies all but stopped spying on Hamas and other violent Palestinian groups in the years after the Sept. 11, 2001, terrorist attacks on the U.S., instead directing resources to the hunt for the leaders of al Qaeda and, later, Islamic State, according to U.S. officials familiar with the shift.

Calculating that Hamas had never directly threatened the U.S. and burdened with other spying priorities, Washington ceded the responsibility to Israel, confident that its aggressive security services would detect any threat, the U.S. officials said. It should have been “a well-placed bet,” said one senior counterterrorism official.