Billionaire Elon Musk bade farewell to Donald Trump in an extraordinary Oval Office appearance, sporting a black eye, brushing aside drug abuse claims and vowing to stay a "friend and adviser" to the US president.

As the world's richest person bowed out of his role as Trump's cost-cutter-in-chief, the Republican hailed Musk's "incredible service" on Friday and handed him a golden key to the White House.

But Trump insisted that Musk was "really not leaving" after a turbulent four months in which his Department of Government Efficiency (Doge) cut tens of thousands of jobs, shuttered whole agencies and slashed foreign aid.

"He's going to be back and forth," said Trump, showering praise on the tech tycoon for what he called the "most sweeping and consequential government reform program in generations".

South-African born Musk, wearing a black T-shirt with the word "Dogefather" in white lettering and a black Doge baseball cap, said many of the US$1 trillion savings he promised would take time to bear fruit.

"I look forward to continuing to be a friend and adviser to the president," he said.

But many people were more interested in the livid black bruise around Musk's right eye.

Speculation about the cause was further fueled by accusations in the New York Times on Friday that Musk used so much of the drug ketamine on the 2024 campaign trail that he developed bladder problems.

The SpaceX and Tesla magnate said that his son was to blame for the injury.

"I was just horsing around with lil' X, and I said, 'go ahead punch me in the face,'" 53-year-old Musk said. "And he did. Turns out even a five-year-old punching you in the face actually is..." he added, before tailing off.

Musk, however, dodged a question about the drug allegations.

The New York Times said Musk, the biggest donor to Trump's 2024 election campaign, also took ecstasy and psychoactive mushrooms and traveled with a pill box last year.

Musk has previously admitted to taking ketamine, saying he was prescribed it to treat a "negative frame of mind" and suggesting his use of drugs benefited his work.

Musk's companies have suffered.

Tesla shareholders called for him to return to work as sales slumped and protests targeted the electric vehicle maker, while SpaceX had a series of fiery rocket failures. (AFP)