“People were troubled,” said Annie Donovan, who previously ran the Treasury office in charge of designating areas as opportunity zones. She and two of her former colleagues said they were upset that the Treasury secretary was intervening to bend rules, though they said they didn’t realize at the time that Mr. Mnuchin’s friend stood to profit. The agency’s employees, Ms. Donovan said, “were put in a position where they had to compromise the integrity of the process.”

The opportunity zone initiative, tucked into the tax cut bill that Mr. Trump signed into law in 2017, has become one of the White House’s signature initiatives. It allows investors to delay or avoid taxes on capital gains by putting money in projects or companies in more than 8,700 federally designated opportunity zones. Mr. Trump has boasted that it will revitalize downtrodden neighborhoods.

But the incentive, also championed by some prominent Democrats, has been dogged by criticism that it is a gift to wealthy investors and real estate developers. From the start, the tax break targeted people with capital gains, the vast majority of which are held by the very richest investors. The Treasury permitted opportunity zones to encompass not only poor communities but some adjacent affluent neighborhoods. Much of the money so far has flowed to those wealthier areas, including many projects that were planned long before the new law was enacted.

Investors and others — including Mr. Milken’s institute — have been pushing the Treasury Department to write the rules governing opportunity zones in ways that would make it easier to qualify for the tax break. That campaign worked, and Mr. Milken is among the potential beneficiaries.

Geoffrey Moore, a spokesman for Mr. Milken, confirmed that Mr. Milken had investments inside opportunity zones, though they are a sliver of his overall real estate holdings. He disputed that Mr. Milken had used his institute or Washington connections to benefit his investments and said no one at the institute “has any specific knowledge of Mike’s personal investments.”

Mr. Moore added that Mr. Milken’s support for opportunity zones was based on his longstanding belief “that jobs and the democratization of ownership are the keys to helping people in economically struggling areas.”

A spokesman for the Milken Institute, Geoffrey Baum, said that “to suggest that the work of the Milken Institute is motivated by or connected to Mr. Milken’s investments is flat-out wrong.” He said the institute advocated changes that were intended to spread the benefits to more low-income communities, not to help the wealthy.