In response, Mr. Musk fired off an epithet-laden barrage at Mr. Duffy, including several posts in which he claimed that Mr. Duffy had below-average intelligence. “The person responsible for America’s space program can’t have a 2 digit IQ,” he wrote.

Mr. Duffy later responded to Mr. Musk’s insults with a social media post that quipped: “Love the passion,” adding that “great companies shouldn’t be afraid of a challenge.”

In a statement, NASA’s press secretary, Bethany Stevens, said Mr. Duffy “has focused the agency on one clear goal — making sure America gets back to the moon before China.”

Mr. Musk did not respond to emailed requests for comment.

Mr. Musk’s targeting of Mr. Duffy is just the latest public attack he has levied against members of the Trump administration and G.O.P. leaders since leaving his government post earlier this year.

In June, just days after departing, Mr. Musk took several shots at President Trump’s signature domestic policy bill, calling it a “disgusting abomination” and condemning House Republicans who voted for it for “making America bankrupt.”

He later suggested the president was covering up details of the Jeffrey Epstein case, claiming without providing evidence that Mr. Trump was named in the investigative files.

Mr. Musk and Mr. Trump appeared to have a rapprochement last month when they were seen shaking hands and sitting next to each other talking at an Arizona memorial for Charlie Kirk, the slain conservative activist.

Mr. Musk has a lot on the line before Mr. Duffy’s agencies.

As both secretary of transportation and acting NASA administrator, Mr. Duffy has authority over sectors that are crucial to some of Mr. Musk’s largest investments, such as space tech, as well as the highways upon which the billionaire wants his self-driving cars to be able to operate.

The Artemis III agreement with NASA, a multibillion-dollar contract awarded to SpaceX in April 2021, envisions a lunar landing by 2027, but NASA officials believe the mission date will slip to at least the following year. Such delays could cost the United States its ability to have an edge in what officials call the “second space race” against China, which has plans for a lunar landing of its own by 2030.

While Mr. Duffy stated on Fox that he is “in the process of opening that contract up,” SpaceX has not yet lost the Artemis III contract.

Instead, Mr. Musk’s company and Blue Origin, the space company owned by another billionaire, Jeff Bezos, were ordered to submit plans for accelerating their missions, which are due on Oct. 29, according to a NASA spokeswoman. (Blue Origin has the Artemis V contract, for a later moon landing mission.) NASA has not yet filed the request for information that would open up either of those contracts to bidding from other companies.

The Artemis missions were insulated from the government shutdown, which has otherwise taken its toll on NASA. An estimated 83 percent of the agency’s workers — including contractors — were put on furlough.

But it is unclear what those shutdown reductions portend for the agency, which is currently the subject of intense debate over its structure and leadership.

Mr. Duffy has said that he believes NASA would benefit from having cabinet-level representation, potentially under the Department of Transportation.

Yet there is a growing base of support for reviving the appointment of Jared Isaacman, a billionaire entrepreneur and a close associate of Mr. Musk’s, to lead NASA.

Mr. Isaacman was Mr. Trump’s nominee to run the space agency until May, when the president yanked his bid just days before a Senate confirmation vote he was expected to secure, citing concerns that he had previously donated to Democrats. That decision contributed to the tension between Mr. Musk and the president that preceded the billionaire’s departure from Washington.

In recent days, a group of Republicans, including the far-right influencer Laura Loomer, have been calling for Mr. Trump to reverse his decision and appoint Mr. Isaacman to run NASA, removing the agency from Mr. Duffy’s portfolio.

Elonbros, what happened?