"He said that the fake media, not certain stories, the fake media are an enemy to the country. We don't have a state-run media in this country. That's what they have in dictatorships," Wallace told Priebus on "Fox News Sunday."
Priebus responded by calling "unsourced" stories about turmoil inside Trump's administration "total garbage."
"If you're going to come out with a story that says Russian spies are talking to your campaign, my God," he said. "I mean, I think you should in some cases, or in most cases, actually have a named source."
Priebus argued that the media has not covered Trump's actions during his first month in office as closely as it has covered his notable failures.
"We covered all of that," Wallace interjected. "Here's the problem. When the President says that we're the enemy of the American people, it makes it sound like if you're going against him, you're going against the country."
He compared Trump's response to critical media coverage to President Barack Obama's response.
"You don't get to tell us what to do, Reince! You don't get to tell us what to do any more than Barack Obama did," Wallace said. "I've got to say he never said that we were an enemy of the people."
"He said a lot of things about Fox News, Chris. I think you ought to go check the tape," Priebus said. "He took plenty of shots."
"No, he took the shots, and we didn't like them, and frankly we don't like this either," Wallace said. "But he never went as far as President Trump has, and that's what's concerning, because it seems like he crosses a line when he talks about — that we're an enemy of the people. That is concerning."
In a tweet posted Friday, Trump claimed that the "FAKE NEWS media" is not his personal nemesis but rather "the enemy of the American People."
Priebus said Saturday that the President should be taken "seriously" in his claim.
"I think you should take it seriously," he said. "If the theory is that the press is supposed to be a free forum of information to speak to the American people, I think it ought to be accurate."
Sen. John McCain (R-AZ) warned in an interview aired Sunday that suppressing critical coverage is "how dictators get started."
"If you want to preserve democracy as we know it, you have to have a free and many times adversarial press," he said.
Rep. Adam Schiff (D-CA) said on Sunday that calling the press "an enemy of the people" is "something that you hear tin-pot dictators say."
"It’s not something you’ve ever heard a president of the United States say," he told Jon Karl on ABC's "This Week."