Top Trump officials were again forced to deny the existence of plots to remove the president, two days after reports said the deputy attorney general, Rod Rosenstein, discussed recording conversations with Trump and invoking the 25th amendment.

The 25th amendment provides for the removal of a president deemed unfit for office.

This month, an op ed in the New York Times by an anonymous “senior official” said the amendment had been discussed within the administration. A book by Watergate reporter Bob Woodward detailed efforts by former staff members to stymie Trump on controversial or notionally dangerous policies.

On Sunday, United Nations ambassador Nikki Haley told ABC’s This Week she was “[at the White House] almost every other week, and I can tell you never has anyone talked about the 25th amendment, never has anyone even questioned the president’s mental stability or anything”.

The secretary of state, Mike Pompeo, like Haley preparing for the UN general assembly in New York and key discussions over North Korea, Russia, Iran and other issues, was also forced to address reports about Rosenstein.

“If you can’t be on the team, if you’re not supporting this mission, then maybe you just ought to find something else to do,” he told Fox News Sunday.

Asked if Rosenstein’s reported behaviour would constitute “being on the team”, Pompeo said: “Not remotely.”

I’ve never heard of it, I don’t think that’s a reality at all among all the cabinet members … That’s absurd Nikki Haley

The Times first reported Rosenstein’s alleged conversations, citing sources briefed on memos kept by former FBI deputy director Andrew McCabe. Other outlets followed up, exploring questions over whether Rosenstein had been sarcastic when he reportedly talked about McCabe “wearing a wire” while meeting Trump.

Trump fired McCabe in March, just short of his scheduled retirement. It was announced this week that McCabe will publish a memoir.

On Friday, Rosenstein issued two statements pushing back on the story. In the second, he said he had “never pursued or authorized recording the president and any suggestion that I have ever advocated for the removal of the president is absolutely false”.

Haley told ABC she had “never been part of” discussions of the 25th amendment within the administration, adding: “I’ve never heard of it, I don’t think that’s a reality at all among all the cabinet members. I’ve just never heard that. That’s absurd.”

The Times reporters stood behind their reporting.

Because of the recusal of Attorney General Jeff Sessions, Rosenstein oversees Robert Mueller, the special counsel investigating Russian election interference, links between Trump and Moscow and potential obstruction of justice by the president.

If Trump were to fire Rosenstein it would be widely seen as a move against Mueller – whose work Haley listed among “distractions” from the administration’s record on the economy and other issues.

As it was reported on Sunday that aides were counseling the president not to fire the deputy attorney general, key allies from outside the White House appeared on Fox News, Trump’s favourite channel.

Senator Lindsey Graham of South Carolina told Fox News Sunday: “He shouldn’t fire Rosenstein unless you believe Rosenstein’s lying … but there’s a bureaucratic coup against Trump being discovered here”.

California congressman and House intelligence chairman Devin Nunes told Fox & Friends that Rosenstein “should have his fair day in court”. But he added: “If Rod Rosenstein really did talk about wearing a wire, then, for sure, he should be fired.”

Trump was at his Bedminster golf club in New Jersey. His Twitter account remained quiet.