Mr Musk has reportedly questioned the influence that Mr Epshteyn has had on appointments.
According to Axios this culminated in their clash in what one source called a “massive blow up”.
At one point Mr Musk accused Mr Epshteyn of leaking details of the transition picks.
“You don’t know what you’re talking about,” Mr Epshteyn reportedly snapped back.
Mr Musk has not commented on the reports, but the Trump-Vance campaign gave its whole-hearted backing to the entrepreneur.
“Elon Musk and President Trump are great friends and brilliant leaders working together to Make America Great Again. Elon Musk is a once-in-a-generation business leader and our federal bureaucracy will certainly benefit from his ideas and efficiency,” a team Trump spokesman said in a statement.
A key point of the feuding is the position of treasury secretary, arguably the most important portfolio in the second Trump administration, tasked with delivering tax cuts.
Mr Musk has reportedly thrown his weight behind Howard Lutnick, saying that he would “actually enact change” if he took up the post.
The alternative to Mr Lutnick, who is chief executive of Cantor Fitzgerald, the financial services company which lost 658 employees in the September 11 attacks, is Scott Bessent, a former money manager for George Soros — a frequent target for conservatives.
According to The New York Times, the battle between the two men is “like a knife fight”.
