NEW YORK -- U.S. President Donald Trump tweeted a critique of the media for being "rude to my very hard-working representatives" on Monday, only minutes after counsellor Kellyanne Conway completed a series of interviews on television morning shows.

Conway's interviews, including one that appeared to signal a thaw in the administration's relationship with CNN, were at times combative, exasperating and fascinating -- an illustration of how the administration and reporters are often talking past each other and how she's become something of a cult figure.

Conway spoke on NBC's "Today" show and ABC's "Good Morning America." Her longest interview, and the one right before Trump's tweet, was with Chris Cuomo on CNN's "New Day." Trump has been particularly critical of CNN during his presidency, and for several weeks the administration had been declining to offer its representatives for interviews on the network.

Her appearances followed a weekend interview with The Record newspaper in New Jersey in which Conway had suggested that there were other ways to conduct surveillance that didn't include tapping phones. The president tweeted without evidence more than a week ago that his predecessor, Barack Obama, had tapped his phone line at Trump Tower.

Conway backed away Monday from any suggestion that alternative methods of surveillance were at play with Obama.

"I'm not in the job of having evidence," she said. "That's what investigations are for."

The White House appeared to soften on Trump's assertion on Monday afternoon when press secretary Sean Spicer said the president was broadly referring to surveillance.

On CNN, Cuomo suggested that Conway was injecting more doubt into a situation that Trump could easily clear up. "It doesn't seem right, Kellyanne," he said.

Conway said that she was "allowed to say things in the news without your questioning anybody's personal integrity." Doubts may have been apparent "maybe to you, and maybe to other people who don't necessarily want Donald Trump to be president."

Cuomo said Conway wasn't being fair.

"My question to you, my question on his baseless claim about wiretapping is not about not wanting him to be president," he said. "That's unfair, and it's hurtful because you are feeding people's animosity."

He asked, "Do you think I'm asking the question because I don't want Trump to be president?"

When she answered no, Cuomo said, "then why put the suggestion out there?"

Moving on to a discussion about health care and why some people are seeing increases in their insurance costs, Conway said that "the biggest reason of all is the disaster of Obamacare and you know it."

When Cuomo said, "C'mon, that's just a slogan," Conway leapt at him.

"You just told millions of Americans who don't have health care that they're just a slogan," she said.

"No, I didn't," Cuomo said. "I did not just tell anybody that they're just a slogan. That's what gets you in trouble. I am not someone who doesn't want President Trump to be president so I'm asking tough questions."

Retorted Conway: "You take it way too personally."

Her interviews were a master class in misdirection, seasoned with agitation about the press' treatment of the administration, an attitude echoed by the president in his Monday tweet. When Cuomo suggested she was creating animosity, Conway noted that she had 24-hour Secret Service protection. She created a social media moment by saying she was "not Inspector Gadget" during a discussion of surveillance methods.

Like Spicer, Conway has already been spoofed on "Saturday Night Live," when Kate McKinnon portrayed her in a "Fatal Attraction"-type relationship with CNN's Jake Tapper. Her comment last month about a nonexistent "Bowling Green" massacre made her a subject of derision. She's been the subject of newspaper articles about whether sexism is involved in criticism of her work.

In media circles, New York University professor Jay Rosen ignited a debate about whether the journalistic rationale for interviewing her had evaporated. At least one program, MSNBC's "Morning Joe," has said it would no longer have Conway on, a promise host Mika Brzezinski repeated on Monday. A petition on the liberal Daily Kos website, with more than 150,000 signatures on Monday, called on the media to stop interviewing her.

Things played differently on the other side of the political spectrum, where the conservative website Newsbusters criticized Cuomo for "whining" in their interview and Conway was praised for being spirited in fighting back.

On the "Today" show, Matt Lauer and Savannah Guthrie tried three times to ask Conway why Trump hasn't offered proof of his claim that he was the victim of an Obama-ordered wiretap. Conway said the truth of the claims would emerge during a congressional investigation, and wouldn't comment on conversations Trump had with his attorney general and other officials. She batted away Lauer's comparison to Trump's discredited birther claims.

Finally, it was Guthrie's turn to become frustrated.

"The media did not bring this up," she said. "President Trump did ... It's not like something a blogger wrote. In fairness, the media are trying to find out about the veracity of his claims."

Conway shifted the subject to health care, and his interviewers followed.

Later, they asked about Friday's favourable jobs report -- noting it was the same monthly set of statistics that Trump had questioned when they relayed positive news about the economy during Obama's presidency, but now touted when he was president. It's an issue of credibility, Guthrie said; Conway said the credibility is in the jobs created.

"In fairness, that doesn't answer the question," Guthrie said.

Conway responded that, "I'm talking about the numbers that do matter to the American public right now."