Elon Musk’s young son has become an internet sensation after apparently telling Donald Trump to “shush your mouth” during at extraordinary outing at the White House.

The astonishing incident came as four-year-old Æ A-Xii  joined his tech billionaire dad as the US President signed executive orders and took questions from the media in the Oval Office on Tuesday (local time).

Musk, the world’s richest man and a close Trump confidante, defended his role as an unelected official who has been granted unprecedented authority to dismantle parts of the US government.

“You can’t have an autonomous federal bureaucracy. You have to have one that’s responsive to the people,” Musk said.

The father of 12 paraded his young son, better known as X, as Trump signed an executive order to further bolster Musk’s Department of Government Efficiency’s powers.

Then came the moment that has been shared across social media.

“I want you to shush your mouth,” X appears to have whispered to Trump, who did not respond.

X was also seen mimicking Musk’s gestures and picking his nose before wiping it on the Resolute desk, gifted to President Rutherford Hayes by Queen Victoria in 1880.

“Trump seethes as President Musk’s kid eats his boogers then paws his desk,” one social media user noted.

The little boy, who is one of three children Musk shares with singer Grimes, also stuck his fingers in his father’s ears – spurring laughter from the assembled journalists.

Grimes apparently learned of her son’s White House appearance on social media, slamming her former partner for parading him in public despite her attempts to keep the child’s identity private.

“He should not be in public like this,” the Genesis singer wrote after a fan alerted her to the footage.

“I did not see this, thank u for alerting me.

“But I’m glad he was polite. Sigh.”

More serious matters

Musk, the Tesla CEO and owner of X, pushed back at criticism that he and his DOGE team have operated largely in secrecy.

“I fully expect to be scrutinised and get, you know, a daily proctology exam, basically,” Musk said.

“It’s not like I think I can get away with something.”

DOGE has provided no information on whom it employs, where it is operating or what actions it is taking inside government agencies. It posts few actual results from its work, providing only dollar figures for purported cuts in specific agencies and little specific detail.

Tuesday’s executive order was the latest effort by Trump and Musk to shrink and align the US government with Trump’s policy priorities. There have already been large-scale buyout offers, attempts to strip civil-service protections from federal workers and the effective shuttering of some federal agencies.

The order sets forth rules requiring government agencies to hire no more than one employee for every four workers who leave, and compels agencies to work with Musk’s team to identify large-scale staff reductions and determine which agency components may be eliminated outright.

Employees whose work is critical to national security, public safety, law enforcement and immigration enforcement are exempt from the cuts.

The push toward mass layoffs comes after the Trump administration attempted to cajole federal workers into accepting buyout offers. That effort has been blocked by a federal judge.

Musk pledged on Tuesday to find $1 trillion in savings through his efforts to identify fraud and waste in the government, a figure that would represent almost 15 per cent of total federal spending.

Trump resisted the suggestion by Democrats and other critics that Musk’s role presents a conflict of interest.

As CEO of rocket maker SpaceX, Musk oversees the company’s contracts with the Pentagon and intelligence community that are worth billions of dollars.

“You know, [if] we thought that, we would not let him do that segment or look in that area, if we thought there was a lack of transparency or a conflict of interest,” Trump said.

-with AAP