The debate, according to officials familiar with the deliberations, centers on whether the United States may take aim at only a handful of high-level leaders of militant groups who are personally linked to plots to attack the United States or whether it may also attack the thousands of low-level foot soldiers focused on parochial concerns: controlling the essentially ungoverned lands near the Gulf of Aden, which separates the countries.

The dispute over limits on the use of lethal force in the region — whether from drone strikes, cruise missiles or commando raids — has divided the State Department and the Pentagon for months, although to date it remains a merely theoretical disagreement. Current administration policy is to attack only “high-value individuals” in the region, as it has tried to do about a dozen times.