The move would also grant a C.I.A. push for permission to expand its program of covert drone strikes, which has included occasional attacks in Yemen and Syria but has largely centered on the tribal region of Pakistan, to Afghanistan — until now the exclusive purview of the military.

A cabinet-level committee of the top leaders of national-security agencies and departments approved the proposed new rules — called the P.S.P., for “Principles, Standards and Procedures” — at a meeting on Sept. 14 and sent the document to Mr. Trump, the officials said. They spoke on the condition of anonymity to describe sensitive discussions about a policy that is not yet final or public. A spokesman for the National Security Council did not contest their account but declined to comment.

One senior administration official described the proposed changes as primarily aimed at making much of the “bureaucracy” created by Mr. Obama’s 2013 rules, the Presidential Policy Guidance, or P.P.G., “disappear.”

The official argued that the replacement rules should be seen as similar to Mr. Obama’s but clearer and less bureaucratic — meaning drone operators and commanders would face fewer internal hurdles to launching specific strikes or raids.

By clearing the way to target rank-and-file Islamist insurgents even without the presence of a high-level leader focused on attacking Americans, the new approach would appear to remove some obstacles for possible strikes in countries where Qaeda- or Islamic State-linked militants are operating, from Nigeria to the Philippines.

However, the new plan would still require higher-level approval to start conducting strikes or raids in new countries under “country plans” that would be reviewed every 12 months. And under international law, the United States would probably also still need to obtain consent from a country’s leaders to use force on their soil to strike at lower-level militants who pose no direct threat to the United States, weakening any self-defense argument.

Even before Mr. Obama left office, the evolving terrorism threat put pressure on the limits that were imposed in 2013. At the time, Al Qaeda was still reeling from the killing of Osama bin Laden, combat troops had left Iraq and were being reduced in Afghanistan, and operations outside war theaters seemed destined to be limited to occasional airstrikes aimed at individual “high-value targets” in Pakistan and Yemen, such as Qaeda leaders.