Former Manchester United defender Stam leaves with Cincinnati second-bottom of the Eastern Conference having only won four games out of 25 this season.
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Cincinnati (4-13-8) has lost three consecutive games and has just one win in its last 16, dating to a 2-0 win over Toronto on June 26.
Marshall, a two-time MLS Cup winner with LA Galaxy, will take charge for the first time as Cincy faces Toronto, the only side below it, Wednesday.
"During our nearly two-month process to identify the next general manager for FC Cincinnati, it was apparent that a head coach change was necessary. Therefore, after a great deal of deliberation, the club has made the decision to now move to the future with Jaap no longer serving as head coach," FC Cincinnati president Jeff Berding said.
"We believe a change in leadership is in the best interest of the club at this time, and a new general manager will lead the search for a new head coach. We are an ambitious club, and we remain focused on achieving our goals, the first of which is making Cincinnati a championship city. We thank Jaap for everything he has done for FC Cincinnati during his time here, and we wish him the best as he can now head home to be with his family."
Said Bakkati and Yoann Damet, who were part of Stam's coaching staff, have also left the club.