It was a remarkable feat. On a day when Democratic members of Congress joined protesters outside the government’s largest humanitarian aid agency and the U.S. defense secretary vowed to use active-duty troops to help stem migrant crossings, it was Gov. Philip D. Murphy of New Jersey who managed to be a primary focus of ire on Fox News.

“I think the governor is pretty foolish saying what he said,” Thomas D. Homan, President Trump’s border czar, told the host Sean Hannity late Monday.

Then Mr. Homan threatened to prosecute Mr. Murphy, who days earlier had seemed to imply that he was housing an undocumented immigrant in a garage apartment on his luxe waterfront estate in Middletown, N.J.

“Won’t let it go,” Mr. Homan promised. “We’ll look into it.”

If Mr. Murphy was, indeed, “concealing an illegal alien,” he said, “I will seek prosecution.”

The warning stemmed from odd and misleading comments Mr. Murphy made Saturday during a freewheeling discussion before an audience at a New Jersey college. The conversation with a leader of a left-leaning political group was streamed live and posted to YouTube.

“There’s someone in our broader universe whose immigration status is not yet at the point that they are trying to get it to,” Mr. Murphy said. “And we said, ‘You know what? Let’s have her live at our house above our garage.’