The board is expected to elect Bennack to the position at its annual meeting in June.
Bennack, 72, is the former president and CEO of the Hearst Corporation, and remains vice chairman of the board of the publishing and broadcasting giant. He has been a member of the Lincoln Center board since 1994, and a vice chairman since 1999.
Crawford has served in the post since 2002, when Beverly Sills stepped down. Bennack was reportedly a candidate to replace Sills three years ago, but declined to be considered.
The most pressing business facing Bennack will be Lincoln Center's long-planned redevelopment, which will start next year with an overhaul of 65th street and the Lincoln Center buildings surrounding it. He may also have to face the departure of New York City Opera, which is dissatisfied with its home at the New York State Theater and is considering building a new venue.
"Frank's long experience with and love of Lincoln Center, as well as his leadership at one of the largest and most successful media companies in the world, make him a superb choice for chairman," said Crawford in a statement. "This is a dynamic time in our history as we move forward in revitalizing the campus."