Politico reported the call on Friday could ease the feuding after an extraordinary day of hostilities - largely conducted over social media - that marked a stark end to a close alliance.
The verbal punches erupted on Thursday after Trump criticised Musk in the Oval Office and the pair then traded barbs on their social media platforms: Trump's Truth Social and Musk's X.
"The easiest way to save money in our Budget, Billions and Billions of Dollars, is to terminate Elon's Governmental Subsidies and Contracts," Trump posted on Truth Social.
"I was always surprised that Biden didn't do it!"
Shares in Musk's Tesla closed down over 14 per cent on Thursday, losing about $US150 billion in market value in the largest single-day decline in value in its history.
Minutes after the closing bell, Musk replied, "Yes," to a post on X saying Trump should be impeached. Trump's Republicans hold majorities in both chambers of Congress and are highly unlikely to impeach him.
Musk earlier said on X that Trump is mentioned in still-secret Justice Department files related to Jeffrey Epstein and he suggested that was why the records have not been released.
Musk provided no support for the claim, but it came amid a spectacular and public disintegration of his once-close relationship with Trump.
The trouble between the two started brewing days ago, when Musk denounced Trump's sweeping tax-cut and spending bill. The president initially held his tongue while Musk campaigned to torpedo the bill, saying it would add too much to the nation's $US36.2 trillion ($A55.6 trillion) in debt.
Trump broke his silence on Thursday, telling reporters in the Oval Office he was "very disappointed" in Musk.
"Look, Elon and I had a great relationship. I don't know if we will anymore," Trump said.
While Trump spoke, Musk responded with increasingly acerbic posts on X.
"Without me, Trump would have lost the election," wrote Musk, who spent nearly $US300 million ($A460 million) backing Trump and other Republicans in the 2024 election.
"Such ingratitude."
In another post, Musk asserted that Trump's signature tariffs would push the United States into a recession later in 2025.
The feud was not entirely unexpected. Trump and Musk are both political pugilists with sizeable egos and a penchant for using social media to punch back against their perceived enemies, and many observers had predicted an eventual falling out.
Even before Musk's departure from the administration last week, his influence had waned following a series of clashes with cabinet members over his cuts to their agencies.
For Trump, the fight was the first major rift he has had with a top adviser since taking office for a second time, after his first term was marked by numerous blow-ups
After serving as the biggest Republican donor in the 2024 campaign season, Musk became one of Trump's most visible advisers as head of the Department of Government Efficiency, which mounted a sweeping and controversial effort to downsize the federal workforce and slash spending.
Musk was frequently present at the White House and made multiple appearances on Capitol Hill, sometimes carrying his young son.
Only six days before Thursday's blow-up, Trump and Musk held an appearance in the Oval Office where Trump praised Musk's government service and both men promised to continue working together.
with AP
