Mr. Trump fired back on Twitter that it had been Mr. Scarborough who asked him to quash the story. “I said no!” the president wrote.
The bizarre back-and-forth, which has drawn blanket coverage on cable news, left Washington veterans stupefied. “I don’t remember anything quite like this,” said Vin Weber, a Republican lobbyist and former congressman from Minnesota.
But the episode also underscores several truths about a president who has yet to shed the habits of his pre-political life. After five months in office, Mr. Trump still kibitzes with television stars, keeps close tabs on his news coverage, and is unafraid to lash out in public against journalists he deems disloyal.
In his years as a real estate developer and socialite, Mr. Trump leaned on gossip outlets to burnish his image — and place news items to harm his enemies. Mr. Scarborough’s account on Friday raised the question of whether Mr. Trump has brought those tactics into the West Wing, using The Enquirer and its wealthy publisher, the president’s close friend David Pecker, as a collaborator.
On the air Friday, Mr. Scarborough said that three White House aides had repeatedly “begged” him to apologize to Mr. Trump for his critical commentary, so that the president would stop the Enquirer story from running.
Allies of Mr. Trump, who has fumed over the often-scathing coverage of him on “Morning Joe,” rejected that interpretation of the exchanges, saying that no ultimatum was issued.
The exact nature of the discussions remained murky. But at one point, Mr. Scarborough contacted Jared Kushner, the president’s son-in-law and one of his top advisers, after learning that The Enquirer was pursuing a story about him and Ms. Brzezinski.
Mr. Kushner told the host that he needed to speak with Mr. Trump about his concerns, according to three people familiar with the conversation who requested anonymity to discuss a private conversation.
When Mr. Scarborough, who has taken to questioning Mr. Trump’s mental fitness on-air, noted that he and the president were not on good terms, Mr. Kushner suggested an apology may be in order, the people said.
On Friday, Mr. Scarborough said that he had text messages and emails to back up his account, but he declined to release them, citing the privacy of his sources.
For its part, the Enquirer — which endorsed Mr. Trump for president and often runs stories favorable to him and critical of his opponents — said it wanted nothing to do with the fight. “We have no knowledge of any discussions between the White House and Joe and Mika about our story, and absolutely no involvement in those discussions,” its editor, Dylan Howard, wrote in a statement.
The story in question was published in early June, under the headline “Joe & Mika: TV Couple’s Sleazy Cheating Scandal.’’ Ms. Brzezinski said in a phone interview that she was rattled when her children and friends received calls from Enquirer reporters before it ran. But she added, “We never once considered calling Donald, let’s just put it that way.”
The Enquirer incident came to light as the “Morning Joe” stars reacted to a furor that broke out on Thursday, when Mr. Trump tweeted that Mr. Scarborough was a “psycho” and claimed he had once seen Ms. Brzezinski “bleeding badly from a face-lift.”
His comments were met with a bipartisan backlash, including from Republican lawmakers crucial to his legislative agenda, and revived a broader discussion of the president’s habit of making insulting remarks about women’s appearances.
Ms. Brzezinski and Mr. Scarborough were once friendly with Mr. Trump, visiting him and his family at his Florida estate. But they are now blistering critics, a reversal that Mr. Trump took as an insulting and personal betrayal.
The controversy spilled into Friday, overshadowing the White House’s plans. The hosts wrote an op-ed in The Washington Post. A meeting between the president and his South Korean counterpart, Moon Jae-in, a sober discussion about North Korean aggression, shared coverage on cable news with segments about the Enquirer’s disreputable past and onscreen graphics including “TV Hosts: White House Used Tabloid Story to Threaten Us.”
By nightfall, CNN — which has engaged in its own war of words with Mr. Trump this year — was airing a discussion of whether the White House had veered into the realm of extortion.
Mr. Pecker, The Enquirer’s publisher, was the subject of a recent profile in The New Yorker examining his decades-long friendship with Mr. Trump, which included allegations that he had quashed stories critical of Mr. Trump during the presidential race. (Mr. Pecker denied this.) The Enquirer endorsed Mr. Trump last year and ran questionable articles about his Republican opponents, including a piece that falsely linked Senator Ted Cruz’s father to the Kennedy assassination.
Dick Morris, a former political consultant who now writes for The Enquirer, said in an interview that the criticism of the president on “Morning Joe”’ was a sign of “the media itself becoming Donald Trump’s political opponent.”
“He’s no longer opposing the Democratic Party,” Mr. Morris said. “He’s opposing the media.”
Presidents before Mr. Trump often griped about their coverage, but took a more hands-off approach in responding to journalists. Barack Obama prided himself on ignoring cable news, and George W. Bush preferred his television tuned to ESPN, said Alex Conant, a former spokesman for Mr. Bush.
“Trump is not the first politician to have a thin skin when it comes to bad media coverage,” said Mr. Conant, who led communications for Senator Marco Rubio’s presidential campaign. “It’s his reaction to it that’s so unusual, and counterproductive.”
The talk-show host Dick Cavett, reached by telephone on Friday, recalled his brush with a president’s ire. Richard Nixon was recorded in the Oval Office complaining about Mr. Cavett’s commentary, and asking his chief of staff, H. R. Haldeman, what to do.
“Is there any way we can screw him?” Mr. Nixon can be heard asking of Mr. Cavett.
So would Mr. Cavett offer any advice for Ms. Brzezinski and Mr. Scarborough?
“It’s a considerable honor,” he said, laughing. “I would say, sit back and enjoy it.”