Billionaire Steven A. Cohen is close to an agreement to purchase a majority stake in the New York Mets from Fred and Jeff Wilpon, a move that would give the hedge-fund manager control over a prominent baseball franchise that has disappointed fans and struggled with financial pressures in recent years.

The deal would increase Mr. Cohen’s current minority stake in the team to 80%, according to people close to the matter. As part of the transaction, Fred Wilpon would remain as the controlling owner for the next five years, while...