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(CNN) —  

Acting White House chief of staff Mick Mulvaney says he and President Donald Trump never discussed his potential resignation following a defiant press conference last week in which the aide admitted a quid pro quo between the US and Ukraine.

“After the briefing and all the blowback and the criticism, did you ever offer or think to offer the President your resignation?” Fox News’ Chris Wallace asked Mulvaney on “Fox News Sunday.”

“No, absolutely not,” Mulvaney responded.

“Was that ever discussed?” Wallace asked, to which Mulvaney replied: “Absolutely, positively not.”

“No, I am – listen, I am very happy working there,” Mulvaney said. “Did I have the perfect press conference? No. But again, the facts are on our side.”

Mulvaney has faced intense scrutiny since a headline-grabbing press conference Thursday in which he said, then denied, that Trump held up an aid package to Ukraine in exchange for an investigation into an unsubstantiated theory that Ukraine, not Russia, was responsible for hacking Democratic Party emails in 2016. Mulvaney had faced internal efforts to oust him before House Democrats moved ahead with their impeachment inquiry against the President, multiple sources told CNN.

Top aides, including Trump’s son-in-law and adviser Jared Kushner, were in the process of reaching out to at least two potential replacements for the top West Wing job shortly before House Speaker Nancy Pelosi announced in late September that she would move ahead with an impeachment inquiry.

While Kushner has fielded complaints about Mulvaney, an administration official insists Kushner did not reach out to any potential replacements for the chief of staff job.

Though the efforts did not come to fruition, they underscore the weakness of Mulvaney’s position even before his performance in the briefing room last week. One person familiar with Mulvaney’s thinking said the search came as Mulvaney himself was looking for an exit after ten months in the role, though people close to Mulvaney have denied he wanted to leave.

A source familiar with Trump’s thinking told CNN on Sunday that after watching media coverage this weekend about Mulvaney’s Thursday news conference and performance in attempting to try to clean it up, the President is increasingly frustrated with his acting chief of staff’s ability to properly communicate their impeachment strategy.

Specifically, Trump’s concern, according to the source, is that Mulvaney is not transitioning enough to the role he is in now – helping to lead the impeachment inquiry defense from the West Wing.

During his Fox News interview on Sunday, Mulvaney said that although he didn’t “have a chance at the (press conference) to do everything (he) wanted to do,” he’s still confident in his overall job performance.

“I still think I’m doing a pretty good job as the chief of staff and I think the President agrees,” he said.

CNN’s Kristen Holmes, Kaitlan Collins, Dana Bash, Jim Acosta, Gloria Borger and Jeremy Diamond contributed to this report.