"For myself, to get the opportunity to work in America and to work in MLS for a club like FC Cincinnati, it's a dream," Stam said in a news release.
"I've always liked the country and I've always liked the mentality of the people in the States as well in what they're trying to achieve."
Stam, who signed a contract through 2021 with a one-year option for the 2022 season, takes over a Cincinnati side that had a league-worst 6-22-6 record in 2019 and lost their first two games this season before play was halted due to the COVID-19 pandemic.
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The 47-year-old Stam, who spent three years with Manchester United and helped them win the 1999 Champions League before joining Italian clubs Lazio and AC Milan, arrives in Cincinnati after a short coaching stint at Feyenoord.
Stam stepped down as Feyenoord manager in October after less than five months in charge with the Dutch top-flight club, who won three of their 11 league games under his leadership.
"We are very happy and pleased to have Jaap Stam become FC Cincinnati's new head coach," Cincinnati General Manager Gerard Nijkamp said.
"He has an extensive background in being part of some of the most successful clubs in the world as a player and has also proven that he can guide teams through transitionary periods as a head coach."