The White House memo seeks to expand the mission of the troops at the border to also include duties such as crowd control and temporary detention. But Mr. Mattis said it left the final decision on what American soldiers and Marines could do — and could not — to the defense secretary.

Mr. Mattis mentioned one possible instance in which troops might act: defending a border agent who was hit by a rock, and detaining the migrant who threw it. But asked whether such a situation might call for the American soldier to be armed, the defense secretary unequivocally said it would not.

“No,” he said. “Not with a firearm.”