In three seasons for Les Violets, the 27-year-old has scored 16 goals and been a regular in the team.
However, Toulouse’s relegation to Ligue 2 means they are having to reduce their salary bill.
Genk’s main target is Heracles forward Cyriel Dessers, but they are also keen to add Sanogo to their ranks.
La Depeche reports that Greek club Olympiacos could be his next destination.
Goalkeeper Baptiste Reynet could also be exiting the Stadium de Toulouse with Nimes and Guingamp interested.