In a live-streamed conversation on X with former President Donald J. Trump last month, Elon Musk raised the idea of a “government efficiency commission.” Such a council could ensure that taxpayers’ money was “spent in a good way,” Mr. Musk said during their more-than-two-hour talk.

“I’d love it,” Mr. Trump replied.

On Thursday, Mr. Trump followed up on Mr. Musk’s suggestion. In a speech at the Economic Club of New York, Mr. Trump said he planned to appoint Mr. Musk, who leads Tesla, SpaceX and X, as the head of a new government efficiency commission if he was elected president in November.

The commission would audit the “entire federal government” and “make recommendations for drastic reforms,” Mr. Trump said.

The announcement capped weeks of discussions between Mr. Trump and Mr. Musk, who have bonded over a mutual hobbyhorse of eliminating wasteful spending, three people close to the men said. Mr. Trump has told Mr. Musk that he wants the tech entrepreneur to slash the federal government’s costs, just as he did at his social media company X, one of the people said.

Mr. Musk, the world’s richest man, has started to influence Mr. Trump as the Republican presidential nominee heads into November’s election. That the tech mogul has the ear of the candidate throws open what each can get out of an alliance — and creates a potential minefield. Their political friendship is particularly tricky because Mr. Musk has a sprawling set of businesses, which may present conflicts of interest if Mr. Trump is elected.

Notably, Mr. Musk’s rocket maker, SpaceX, and his electric automaker, Tesla, have received billions of dollars in federal contracts and subsidies. Mr. Musk and some of his companies are also under scrutiny for various infractions from federal agencies including the Department of Justice, the Federal Trade Commission and the Securities and Exchange Commission.