A shy father of seven, Mr. Cohen long ruled his multibillion-dollar hedge fund from a workstation in the middle of a sprawling suburban Connecticut trading floor. Bespectacled and favoring half-zip fleeces, Mr. Cohen has long been known as a workaholic, holding investment meetings on Sunday evenings and personally weighing in on big trades. In recent years, he began to groom his next generation of portfolio managers and became more visibly involved in philanthropy, donating tens of millions of dollars to benefit the mental health of veterans and giving $50 million to the Museum of Modern Art.

Mr. Cohen’s first troubles with insider trading allegations occurred in 1986, when officials from the Securities and Exchange Commission questioned him about his trades in shares of RCA ahead of an announcement that RCA would be bought by General Electric. During the interview with S.E.C. officials, he invoked his Fifth Amendment right against self incrimination several times, according to a Reuters report, which cited a transcript of the deposition. His ex-wife, Patricia Cohen, later claimed in a lawsuit that he had used proceeds from those trades to start S.A.C. Capital.

Federal prosecutors, foremost among them Preet Bharara, then the United States attorney for the Southern District of New York, combed through the records of Mr. Cohen’s fund two decades later, hoping to catch him in an illegal act. They wiretapped his phones, but he quickly found out about the surveillance.

He became a version of the white whale to Mr. Bharara’s Ahab, and in 2013, federal prosecutors filed criminal charges against S.A.C. — not Mr. Cohen personally. He paid $1.8 billion to settle the charges and was not able to manage other people’s money from 2016 to ’18. After selling off a chunk of his giant art collection for $88 million and changing his Greenwich, Conn., firm’s name to Point72 Asset Management, Mr. Cohen retreated.

Until recently. His ban on managing outside funds expired last year. He had already reopened a shuttered London office. He battled a sex discrimination lawsuit against his firm, which was accused of underpaying female employees and condoning crude behavior toward women. The case was sent to private arbitration.

Now, seven years after he pursued a purchase of the Los Angeles Dodgers and paid $20 million for a small share of the Mets, Mr. Cohen is on the verge of gaining the kind of exposure that comes only with owning a major sports franchise in a world capital at a time when team owners, such as Mark Cuban, are among the country’s biggest celebrities.

Mr. Cohen currently owns 8 percent of the team, according to one of the people who spoke to The New York Times on condition of anonymity. He purchased his share after the collapse of an announced $200 million sale of a minority stake in the team to the hedge fund manager David Einhorn.