Congress has been wrestling for years about whether and how to update the war authorization law. On its face, the statute permits using force only against those responsible for the Sept. 11, 2001, terrorist attacks. But three presidents, from both major parties, have stretched it to cover enemies beyond the original Al Qaeda and its Taliban host in Afghanistan.

And broader questions about the legal scope and limits of presidential warmaking powers have taken on new urgency in the Trump era, including because of tensions with North Korea over its testing of nuclear weapons and longer-range missiles and the deaths of four American soldiers this month in Niger.

Questioned about North Korea on Monday, Mr. Tillerson and Mr. Mattis acknowledged that Congress had not granted any authority to Mr. Trump to start a war. Still, they said, he possessed constitutional authority as commander in chief to strike North Korea in the event of an imminent attack against the United States.

But Mr. Tillerson demurred when pressed to say whether North Korea’s mere possession of a nuclear weapon that could reach the United States would be sufficient to enable Mr. Trump to strike on his own. Mr. Tillerson said such questions could turn on the facts, including whether such a weapon was stored in a bunker or was sitting on a launching pad.

Several senators also pressed the witnesses about the recent deaths in Niger. Although Mr. Mattis said the purpose of the deployment there was to help build up national defenses because “the enemy is trying to move somewhere” as the Islamic State’s stronghold in Iraq and Syria collapsed, he said it was authorized under a separate law that allowed the military to carry out training and assistance missions.