"Unfortunately, the manner in which Klinsmann took his leave means further co-operation with him on the supervisory board is not conceivable," Hertha's investor Lars Windhorst told a press conference.
"In the club's interest, constructive co-operation can't continue in this way."
The 1990 World Cup winner, who only took charge of Hertha in November, dropped a bombshell on Tuesday when he quit as head coach, three days after a 3-1 home defeat to Mainz pushed them towards the German league's relegation zone.
In a Facebook video, Klinsmann, 55, initially said he wanted to return to his position on the supervisory board, a position he only held for three weeks last autumn before stepping into the head coach role.
However, he later criticised Hertha's general manager Michael Preetz, hinting at a power struggle, which left Windhorst with little choice but to rule out Klinsmann's return in an advisory capacity.