The 47-year-old has signed a two-year contract with Cincinnati, who finished bottom of the Eastern Conference last season.
Cincinnati started the 2020 MLS season with two defeats before the campaign was halted because of the coronavirus pandemic.
Stam had been without a club since resigning as Feyenoord manager in October last year after less than three months in charge of the Eredivisie club.
He has previously managed Reading in the second-tier of English football, and nearly took them to Premier League promotion in 2017 but lost out to Huddersfield on penalties in the play-off final.
"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 told his new club's website.
"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.
"To get the opportunity to work for FC Cincinnati and this project by building the club – hopefully even further than where they already are, because I think they've already done a great job – but being a presence as a team, even more in MLS, getting results and working together with the fans and everybody already working within the club, for me it's a big challenge to do well."
Stam was part of the Manchester United side that famously won the treble in 1999 while he also had spells at Lazio, AC Milan and Ajax Amsterdam.
