Donald Trump has “full faith” in his chief of staff, John Kelly, and is “not actively searching for replacements”, his key aide Kellyanne Conway told CNN on Sunday.

“The president says Gen Kelly is doing a great job,” she said.

The White House budget director, Mick Mulvaney, echoed Conway’s comments in a separate Sunday morning interview, saying Trump and administration officials were “absolutely not” seeking Kelly’s removal.

“I think that talk about the chief’s departure is much ado about nothing,” Mulvaney told Fox News.

The comments amount to a firm rebuttal of speculation that Trump has lost faith in Kelly after his handling of the controversy over Rob Porter, the White House aide who resigned last week after allegations of domestic violence from his ex-wives.

“The president is very disturbed by what he sees, absolutely,” Conway said, asked about Trump’s comments expressing sympathy for Porter – but not his alleged victims – and his tweet on Saturday saying that “people’s lives are being shattered and destroyed by a mere allegation”.

“You can feel that someone did a great job for you ... you can talk about their competence ... and you still feel horrified when you see pictures and contemporaneous reports,” Conway said.

And she criticized political opponents who, she said, were attempting to conflate the Rob Porter issue with the #MeToo movement. The president, she said, “is sympathetic to women and men who are victims of domestic violence”.

Conway said the Porter described in the police reports against him was “not the Rob Porter we knew in the White House”.

“I have no reason not to believe the women, and a week ago I had no reason to believe this had happened,” she said. “Rob Porter did the right thing when he resigned.”

She added that she had no concern for the safety of Hope Hicks, the White House communications director who is reportedly currently dating Porter.

“I don’t worry about her in that respect,” the White House counselor said. “I’ve rarely met someone so strong ... Hope carried on this week, she’s doing her job.”

On Saturday Trump appeared to offer a new defence of Porter, tweeting: “People’s lives are being shattered and destroyed by a mere allegation. Some are true and some are false. Some are old and some are new. There is no recovery for someone falsely accused – life and career are gone. Is there no such thing any longer as Due Process?”

The post was interpreted as a comment not just about his former aide, or a second White House staffer, speechwriter David Sorensen, who was on Friday also forced to resign despite denying similar accusations, but also on the wider #MeToo movement that has encouraged women to speak out since the Harvey Weinstein scandal last year.

The White House legislative director, Marc Shortt, told NBC: “I think the president is shaped by a lot of false accusations against him in the past,” a reference to numerous allegations of sexual misconduct made against Trump, which he denies.

Porter, a key White House aide, left the administration on Wednesday after two ex-wives accused him of physical abuse. The allegations, first reported by the Daily Mail, included a photo of Porter’s first wife with a black eye that she said her ex-husband had given her.

The White House has faced criticism over its response to the allegations against Porter, including an initial defense of the White House aide as a “man of integrity” from the chief of staff, John Kelly. There have also been questions raised about when top White House staffers learned of the allegations, which had blocked Porter from receiving a permanent security clearance.

Porter has dismissed the allegations against him as “outrageous” and “simply false”.

Trump previously defended Porter to reporters on Friday, saying: “We wish him well, he worked very hard. We found out about it recently and I was surprised by it, but we certainly wish him well and it’s a tough time for him.

“He did a very good job when he was in the White House. And we hope he has a wonderful career and he will have a great career ahead of him. But it was very sad when we heard about it and certainly he’s also very sad now.”

He then added a defense of Porter: “He also, as you probably know, says he’s innocent and I think you have to remember that. He said very strongly yesterday that he’s innocent so you have to talk to him about that, but we absolutely wish him well, he did a very good job when he was at the White House.”

The Porter controversy intensified pressure on Kelly over his strong initial defence of the official.

Trump has long been skeptical of allegations by women claiming abuse by men. He endorsed the Alabama Republican Roy Moore for Senate in 2017 despite statements that the Republican hopeful sexually assaulted underage girls, noting that Moore “denies it”.

Trump also has pushed back on at least 19 allegations of sexual misconduct against him by saying that the accusers were all liars. The White House spokesperson Sarah Sanders affirmed this was the official position of the administration in October 2017.

“Those accusers have had their day on your network and elsewhere for a long time. They were tried out again late last year,” Conway told ABC on Sunday.

However, Trump has been more comfortable making allegations about others. He led the charge against the Central Park Five, five African American youths wrongly convicted of rape, running advertisements implying they should be executed.

He has maintained, in spite of DNA evidence, that the five are guilty.