Elon Musk stood before a giant American flag at a Wisconsin political rally in March and rolled out an eye-popping allegation of rampant fraud at the Social Security Administration. Scammers, he said, were making 40 percent of all calls to the agency’s customer service line.

Social Security employees knew the billionaire’s claim had no basis in fact. After journalists followed up, staff members began drafting a response correcting the record.

That’s when Leland Dudek — plucked from a midlevel job only six weeks earlier to run Social Security because of his willingness to cooperate with Mr. Musk’s Department of Government Efficiency — got an angry call from the White House, according to several people familiar with the exchange.

“The number is 40 percent,” insisted Katie Miller, a top administration aide who was working closely with Mr. Musk, according to one of the people familiar with the April 1 call. President Trump believed Mr. Musk, she said. “Do not contradict the president.”

Throughout the early months of this Trump presidency, Mr. Musk and his allies systematically built a false narrative of widespread fraud at the Social Security Administration based on misinterpreted data, using their claims to justify an aggressive effort to gain access to personal information on millions of Americans, a New York Times investigation has found.

Their work has led to the departures of thousands of employees, thinning an already overstretched work force and setting off a wave of public anxiety over the state of an agency administering politically sacrosanct retirement benefits that Mr. Trump has vowed to protect.