Genesio, who has previously managed Lyon, Rennes and Lille, was expected to take over at the Stade Velodrome ahead of the new season.
"I chose to join Olympique de Marseille because I was drawn to the challenge presented to me," Genesio said in a club statement.
"Olympique de Marseille is a unique club, with an exceptional history, a strong identity, and supporters whose passion is recognised far beyond its borders. I am very happy to be part of this new project."
The 59-year-old recently left Lille at the end of his two-year contract after overseeing 99 matches in all competitions. He guided the club to a third-place finish in Ligue 1 last season, securing Champions League qualification.
A former midfielder, Genesio played more than 200 matches across the French first and second divisions before moving into coaching.
He began his managerial career with Lyon, taking charge of 185 matches and leading the club to two Champions League qualifications.
Genesio later managed Chinese side Beijing Guoan before joining Stade Rennais, where he spent two years and was named Ligue 1 Manager of the Year in 2022.