Van Bronckhorst, who enjoyed a distinguished playing career with the likes of Rangers, Arsenal and Barcelona, is currently assistant coach at De Kuip, a role he also held under Rutten's predecessor Ronald Koeman.
The 40-year-old is now set to take the reins from Rutten - who announced earlier this month that a disagreement with the Eredivisie club's hierarchy over the team's direction was behind his decision to leave.
Van Bronckhorst started his career with Feyenoord before returning to the club to see out his playing days in a three-year spell between 2007 and 2010.
"I once started [here] as a newbie and after a long career I have returned and now I will become head coach," he told the club's official website.
"As an assistant coach I learned a lot from both Ronald Koeman and Fred Rutten.
"It is good for the club that there is now clarity towards next season."
Van Bronckhorst, who was capped 106 times by Netherlands, won the domestic titles in Scotland, England and Spain and also lifted the UEFA Champions League with Barca in 2006.
