WASHINGTON—The Central Intelligence Agency is racing to deal a death blow to al Qaeda while the network is weakened, raising concerns for some U.S. officials that the campaign of drone strikes could become so politically damaging for Pakistan's leaders that they may seek to curtail them.

U.S. officials said over the weekend that a CIA drone strike killed al Qaeda's second in command on Aug. 22, a major blow to militants in their country and a sign that American officials are...