The announcement followed Indonesia sacking coach Shin Tae-yong on Monday as they battle for a place at the 2026 World Cup.
Kluivert who won the Champions League title as a player with Ajax in 1995, will take charge of Indonesia until 2027, with an option to extend further, the country's football federation said in a statement.
Indonesia are third in their World Cup group after six of 10 matches, a point behind Australia in the battle for the second automatic spot in the finals, which will be expanded from 32 to 48 teams in the contest jointly hosted by the United States, Canada and Mexico.
Since ending his playing career in 2008, Kluivert has held various roles, including coach of Turkish side Adana Demirspor and the Dutch Caribbean island of Curacao.
He has served as director of football for Paris St Germain, assistant coach of the Netherlands and Cameroon, and manager of the youth team of Ajax.
Kluivert will be unveiled in Jakarta on Sunday and will be joined in Indonesia by two assistant coaches, fellow Dutchmen Alex Pastoor and Denny Landzaat.
His first competitive match in charge will be Indonesia's trip to Australia for a World Cup qualifier on March 20.
Indonesia's sole World Cup appearance came during Dutch rule in 1938, as the Dutch East Indies. If they finish third or fourth, they could still reach the 2026 finals through further rounds of qualifying and an intercontinental playoff.