President Donald Trump hands his pen to Director of the Office of Management and Budget (OMB) Mick Mulvaney (front-L), after signing an executive order on March 17, 2017 in Washington, DC.

With John Kelly under fire for his handling of a top aide's domestic abuse scandal, there is a new round of reports that President Donald Trump is entertaining the possibility of replacing the retired four-star Marine general as his chief of staff.

This time, the name popping up as a potential successor is Mick Mulvaney, director of the administration's Office of Management and Budget and the Consumer Financial Protection Bureau.

Mulvaney, a Republican former congressman from South Carolina who positions himself as a fiscal conservative, has seen his star rise in an administration often rocked by chaos and staff turnover. CNN reported that Mulvaney's profile grew even bigger during negotiations over government spending.

Trump has reportedly often chafed at Kelly's rigidity in controlling access to the president. This week, though, questions about the chief of staff's fitness for the job emerged as he staunchly defended a key aide, Rob Porter, who was accused by two ex-wives of physical and emotional abuse.

Kelly also reportedly knew about the allegations for months before reports broke out this week and encouraged Porter to remain in the administration. Porter, who quit the White House staff as the media spotlight grew, has denied the accusations.

On Friday, meanwhile, The Washington Post reported that two senior officials said Kelly told staff to share a version of the Porter timeline which differed from previous accounts. Some aides, the Post added, consider Kelly's version to be untrue.

The New York Times reported that Trump is upset with Kelly for creating unwelcome attention this week as the chief of staff called immigrants "lazy" and defended Porter. Citing two sources, the Times reported that Trump has asked advisors about what they think about Mulvaney possibly becoming chief of staff.

CNN backed up the Times, reporting that the president has been asking aides about their opinion of Mulvaney, although the news network said Trump has been "coy" when seeking aides' thoughts on the matter. Yet, CNN added that most of the advisors assume that Trump is referring to the chief of staff spot.

Despite the reports of tension between the president and his chief of staff, Trump has often praised Kelly in public. White House spokesman Raj Shah told a news briefing Thursday that Kelly still has the confidence of the president. Shah also told reporters Thursday: "I think it's fair to say we all could have done better over the few hours or last few days in dealing with this situation."

Kelly himself succeeded Trump's first chief of staff, Reince Priebus, in late July. Priebus' tenure in the position was racked by constant turmoil and often-competing leaks to the media by staffers at odds with each other. Ironically, the Times reported that Trump recently complained about Kelly to none other than Priebus.

A White House spokeswoman did not immediately respond to CNBC's request for comment about the speculation surrounding Mulvaney.

Gary Cohn, Trump's chief economic advisor, is another name that often comes up as a possible chief of staff candidate. When asked about that, a White House official, who spoke to CNBC on condition of anonymity, called the speculation "premature."

"That said, he could do it and would be great," the official added. "Not sure if he would. But Kelly has been a star."

— CNBC's Eamon Javers contributed to this report.