The president of the HFA Sandor Csanyi said in an interview with sports daily Nemzeti Sport that the board had lost faith with the team's progress under Storck and the German coach agreed his time was up.
"Firstly, I am grateful and honoured to have had a chance to work for a country with great traditions like Hungary. The board and I felt the same, I had to leave," Storck said on the HFA's website.
Storck, 54, was appointed national team coach in July 2015 and led Hungary to their first major tournament appearance since the 1986 World Cup when they qualified for Euro 2016.
Storck's Hungary finished top of their group at the Euros, having beaten Austria and drawn with Iceland and Portugal, but eventually lost to Belgium in the last 16.
They endured a disappointing World Cup qualifying competition, however, and finished third in their group after losing to Andorra and drawing at the Faroe Islands.
Storck managed Hungary for 25 games, winning eight, drawing seven and losing 10 matches.
Hungary have friendlies in November against Luxembourg away, and at home against Costa Rica.
(Reporting by Zoltan Fazekas; Editing by Toby Davis)
