Veh informed club bosses of his decision not to extend his contract beyond June prior to Sunday's 2-1 win over Stuttgart, and he told the players of his choice before that game, according to the Frankfurter Allgemeine newspaper.
Club president Heribert Bruchhagen on Monday confirmed reports to the dpa agency, with the search for a successor now under way.
Veh is understood to have made up his mind after Frankfurt were eliminated from the Europa League by Porto on Thursday.
Sunday's win lifted the Eagles six points clear of the relegation zone and Veh, who led Stuttgart to the Bundesliga title in 2007, will aim to keep the club in the top flight before leaving.
Inevitably, Veh is already being linked with other jobs and he has emerged as the favourite to succeed Jens Keller as coach of Schalke from next season.
Keller is under pressure again after his side were beaten 6-1 by Real Madrid and 5-1 by Bayern Munich in the space of four days.