Trump said on Saturday his relationship with Musk was likely over after they exchanged criticism on social media, with the Tesla and SpaceX CEO describing the president's sweeping tax and spending bill as a "disgusting abomination".
Musk has since deleted some posts critical of Trump, including one signalling support for impeaching the president.
Sources close to the world's richest man say his anger has started to subside and that he may want to repair the relationship.
Company and market analysts suggested Musk's tone could reflect a desire to protect his businesses.
"I regret some of my posts about President Donald Trump last week. They went too far," Musk wrote in a post on his social media platform X on Wednesday, without saying which specific posts he was talking about.
After Musk's comments, Trump told the New York Post: "I thought it was very nice that he did that."
Vice President JD Vance and David Sacks, Trump's crypto czar and a longtime Musk friend, pushed for Musk and Trump to make up, according to two people briefed on the conversations, with one of the sources saying that Musk was serious about potentially starting a third party, a risk for Republicans going into midterms.
Tesla shares rose 1.3 per cent on Wednesday.
"The conciliatory tone from Musk recently might indicate his desire to protect his businesses in the light of the position he has found himself in," said Mamta Valechha, consumer discretionary analyst at Tesla investor Quilter Cheviot.
Tesla shareholder Matthew Britzman, an analyst at Hargreaves Lansdown, said both Musk and Trump appeared to have de-escalated the situation.
"It still feels unlikely that we'll see these two giant personalities so closely intertwined again but it's in neither's best interest to let the drama continue," he said.
Shawn Campbell, adviser and investor at Camelthorn Investments, said the relationship between Musk and Trump could be restored but also said it was unlikely it would return to where it once was.
"The stakes between the richest man in the world and leader of the most powerful nation in the world are just so big, with billions of dollars of government contracts at stake, not to mention the power to investigate and regulate and tax," said Campbell, who personally holds Tesla shares.
Vance and White House Chief of Staff Susie Wiles spoke to Musk on Friday, one of the people briefed said.
That same day, James Fishback, a Tesla investor and external DOGE adviser, posted on X that the billionaire owed Trump "a full-throated apology".
Musk retorted: "What's the apology for exactly."
But Musk subsequently deleted several posts, and a senior Trump adviser called Fishback to thank him, according to one of the sources.
Musk bankrolled a large part of Trump's 2024 presidential campaign, spending nearly $US300 million ($A459 million) in last year's US elections and taking credit for Republicans retaining a majority of seats in the House and retaking a majority in the Senate.
Trump then named him to head an effort to downsize the federal workforce and slash spending.
Musk left the role late last month after criticising Trump's marquee tax bill, calling it too expensive and a measure that would undermine his work at the Department of Government Efficiency.
On Monday, Trump said he would not have a problem if Musk called and that he had no plans to discontinue the Starlink satellite internet provided to the White House by Musk's SpaceX but might move his Tesla off-site.
"We had a good relationship, and I just wish him well," Trump said.
Musk responded with a heart emoji to a video on X showing Trump's remarks.
Tesla shares have recouped all the losses they suffered during the public feuding between Trump and Musk last Thursday, when more than $US150 billion was wiped off the company's market value.
