The 55-year-old, who also managed Midlands rivals Birmingham City and Aston Villa, had been out of the game since leaving Nottingham Forest last year.
Genk are in the lower reaches of the 16-club Belgian first division, although their prospects brightened last night with a 3-2 home win against Ghent. McLeish will be reunited with midfielder Thomas Buffel, formerly of Rangers, when he starts his new role on Monday.
The Belgian club said on its Facebook page: "KRC Genk is proud to present its new trainer: Alex McLeish. He signed a contract for two seasons. Welcome Mr Alex McLeish."
McLeish succeeds Emilio Ferrera, who was sacked after Genk's 3-1 defeat by Mechelen in their opening league fixture. Thibaut Courtois, Kevin de Bruyne and Christian Benteke all started out at the club.