The Trump administration has insisted that its boat strikes are lawful, telling Congress in September that Mr. Trump had “determined” that the United States was in a noninternational armed conflict.

A secret Justice Department memo blessing President Trump’s boat strikes as lawful hangs on the idea that the United States and its allies are legally in a state of armed conflict with drug cartels, a premise that derives heavily from assertions that the White House itself has put forward, according to people who have read it.

The memo from the department’s Office of Legal Counsel, which is said to be more than 40 pages long, signed off on a military campaign that has now killed 80 people in the Caribbean Sea and eastern Pacific Ocean. It said such extrajudicial killings of people suspected of running drugs were lawful as a matter of Mr. Trump’s wartime powers.

In reaching that conclusion, the memo contradicts a broad range of critics, who have rejected the idea that there is any armed conflict and have accused Mr. Trump of illegally ordering the military to commit murders.

The administration has insisted that Mr. Trump has the authority to lawfully order the strikes under the laws of war, but it has provided scant public details about its legal analysis to buttress that conclusion. The accounts of the memo offer a window into how executive branch lawyers signed off on Mr. Trump’s desired course of action, including appearing to have accepted at face value the White House’s version of reality.

The memo, which was completed in late summer, is said to open with a lengthy recitation of claims submitted by the White House, including that drug cartels are intentionally trying to kill Ameri