Jeffrey A. Zucker hasn’t been getting a lot of sleep lately. But he says that’s nothing new.

“I don’t sleep that much anyway,” Mr. Zucker, the president of CNN, said on Wednesday in his fifth-floor office, just off the network’s glassy Midtown Manhattan newsroom.

A television executive with a reputation for pugilism — a plaque above his desk reads “Punch Today in the Face” — Mr. Zucker, 52, has weathered decades of battles in his industry. But he and CNN are in the middle of their most intense bout yet: an unlikely public fight with the leader of the free world.

It is rare that a single news organization attracts the level of ire mustered by President Trump, who over the weekend posted on Twitter a video that portrayed him wrestling a figure with the logo of CNN for a head.

But the president’s denunciations — in stinging tweets and slashing speeches, in phrases like “fraud news” and “garbage journalism” — have far outstripped his criticisms of other prominent news outlets, like The New York Times or The Washington Post. And his attacks have spawned a cottage industry of Trump supporters who have declared a digital war of sorts against CNN, including gotcha videos of network employees and threatening messages sent to anchors’ cellphones.