After a contentious interview on Tuesday in which Kellyanne Conway, the presidential adviser, and Jake Tapper of CNN clashed over whether the White House or the news media was more indifferent to facts, the journalist was bracing on Wednesday for a backlash from the right.

While the 25-minute interview touched on issues like the confirmation of Betsy DeVos as eduction secretary and President Trump’s relationship with President Vladimir V. Putin of Russia, the bulk of it focused on Mr. Trump’s scathing criticism of the media and the administration’s penchant for repeating falsehoods.

The interview was frequently testy, and Mr. Tapper’s performance was widely praised by those seeking a more aggressive approach to fact-checking the Trump administration.

Kellyanne Conway’s full interview with Jake Tapper
Video by CNN

But then Axios, citing an anonymous source, reported that Mr. Tapper would now likely be a target for “hit pieces” from at least one conservative news site. The Times was unable to independently verify the report.

Mr. Tapper, however, appeared to welcome the critics, tweeting that he would just wait for “attack dogs” to come at him as ordered.

Ms. Conway’s CNN appearance came amid growing questions about her credibility after she falsely spoke of a “Bowling Green massacre” that never happened and used the phrase “alternative facts” to justify the administration’s view of the truth. She repeatedly corrected the Bowling Green error and apologized for it on Tuesday, but reports later emerged that she had previously invoked the nonexistent episode in an interview with Cosmopolitan magazine (which did not print the quote) and with TMZ.

Some critics are calling on cable news outlets to bar her from appearing on air, and CNN declined to have her on as a guest Sunday.

Mr. Trump added to tension with the press by accusing news organizations of playing down or declining to cover terrorist attacks. The White House on Monday issued a list of what it said were 78 underreported attacks; but several news organizations, including The New York Times, provided evidence to the contrary.

On Tuesday, Ms. Conway used her airtime to question the focus and tone of more recent news coverage. Without citing examples, she criticized descriptions of Mr. Trump that lacked “a certain respect for and recognition of the dignity for the office of the president.”

She asked for “more complete coverage” of Mr. Trump’s executive orders, instead of a focus on the travel ban affecting seven Muslim-majority countries.

Mr. Tapper countered that the executive order had caused chaos at airports across the globe and was a major story deserving coverage. He added that Mr. Trump has also spread false information that the president accuses the media of not reporting.

In one example cited by Mr. Tapper, Mr. Trump said on Tuesday that the murder rate was “the highest it’s been in 47 years” and that “the press doesn’t tell it like it is.” The facts: While the rate spiked in 2015, multiple fact-checks found that the murder rate is actually near a 45-year low.

“Facts are stubborn things,” Mr. Tapper said, “and to say that we’re not reporting something that happens not to be true, therefore we’re not to be trusted, that’s a problem.”

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When Ms. Conway suggested that the media had played down some attacks because people had become “inured to terrorist attacks,” Mr. Tapper protested.

“Well, your spin about the idea that we don’t want to be inured to that, that’s a lovely spin, but that’s not what he was saying, Kellyanne,” he said. “He was saying the media does not cover these stories because we don’t want to cover them because we have some sort of agenda.

“That’s what he was suggesting, and it’s offensive given the fact that CNN and other media organizations have reporters in danger right now in war zones covering ISIS, and I just don’t understand how the president can make an attack like that.”

On Wednesday, after the report that he would be targeted, Mr. Tapper appeared to be enjoying the #TapperDirtFile Twitter hashtag that had been created in his honor. Users professed to expose damaging, but made-up, tidbits to be used in a hit piece about him, such as liking Nickelback or the time he ordered a “large coffee” at Starbucks.

He even added his own “dirt”: