New York (CNN Business) Several members of the media were shoved and assaulted at President Trump's campaign rally in El Paso, Texas, last week.

But even as some supporters appear to take cues from Trump's anti-media rhetoric, the president isn't changing his tune.

On Wednesday he said "The Press" -- a catchall term for tens of thousands of people -- is "totally out of control."

He singled out The New York Times for its recent reporting about him, calling the newspaper "a true ENEMY OF THE PEOPLE!"

In the past Trump has hurled this insult at groups of news organizations, including CNN, but it's rare to see him point at a single news outlet and call it an "enemy."

A.G. Sulzberger, the publisher of The Times, responded on Wednesday by saying "the phrase 'enemy of the people' is not just false, it's dangerous. It has an ugly history of being wielded by dictators and tyrants who sought to control public information. And it is particularly reckless coming from someone whose office gives him broad powers to fight or imprison the nation's enemies."

Sulzberger recently spoke with Trump about this very issue during an Oval Office interview. Trump responded by claiming that he thinks he is a great defender of the free press.

Trump's newest attack on Wednesday was apparently sparked by a Times investigation into potential obstruction of justice by the president.

Trump initially claimed that the paper's reporters did not contact him for comment.

He followed up with a Wednesday morning tweet that said "writers don't even call asking for verification."

Times reporter Maggie Haberman said "that's a lie."

For this new investigation, "we reached out to them on Friday," Haberman said on CNN's "New Day."

"I sent several e-mails that went unanswered until yesterday. We went through a detailed list of what we were planning on reporting," Haberman said. "They chose not to engage, and afterwards, the president acts surprised. Whether his aides are not telling him what we are looking at, or whether this is a game and he knows what it is and he's pretending that he doesn't -- I can't read his mind. But we certainly follow normal reporting practices, and went over it at length with both the White House and the Department of Justice."

She added: "I find it hard to believe that his staff didn't brief him once again that this kind of a report was coming."

Meeting Trump where he is, on Twitter, The Times sent him a series of tweets on Wednesday about the reporting process.

"The White House was given 5 days to respond to the details of our investigation," the newspaper's public relations team wrote.

Sulzberger's statement dealt more broadly with the consequences of Trump's "enemy" rhetoric.

"As I have repeatedly told President Trump face to face," he said, "there are mounting signs that this incendiary rhetoric is encouraging threats and violence against journalists at home and abroad."

Last week's incident appeared to be another example. A rallygoer in El Paso breached the area reserved for reporters, known as the "press pen," during Trump's speech. According to footage of the incident, he began shoving multiple people, including BBC cameraman Ron Skeans.

"I didn't know what was going on," Skeans told the BBC. He described the contact as a "very hard shove."

The attacker was restrained by another member of the media, pro-Trump radio host and live-streamer Ben Bergquam.

"I was up there filming and I looked to my right. I heard a commotion and I started seeing cameras falling down and people falling over, and I saw a guy running towards me, falling, tumbling towards me, with a MAGA hat on," Bergquam said on Sunday's "Reliable Sources."

Bergquam said he agrees with Trump's "fake news" claims, but that the attacker "made all MAGA-hat wearers look bad."

Trump noticed the disturbance and paused his remarks to make sure everything was OK.

Bergquam restrained the man and handed him off to security personnel.