Stam, 47, takes over the US club following managerial stints at Reading in England and PEC Zwolle and Feyenoord in his native Netherlands.
He stepped down as coach of Feyenoord last October after their 4-0 thrashing at the hands of Ajax, the club saying at the time they wanted him to stay.
Interim manager Yoann Damet has been at the helm of FC Cincinnati this season after coach Ron Jans resigned in February - 12 days before the start of the 2020 campaign - in the wake of a league investigation into an alleged racist remark.
"We are very happy and pleased to have Jaap Stam become FC Cincinnati's new head coach," Nijkamp, who worked with Stam at Zwolle, said. "He has an extensive background in being part of some of the most successful clubs in the world".