The Austrian has been told that Italian rookie Vitantonio Liuzzi, although completing an initial three race stint at Monaco, will stay in the seat this weekend.
Moreover, beyond Montreal, it is not clear whether Klien, 22, is slated to compete at Indianapolis' successive round, or beyond.
"After having evaluated both drivers' performance during their three race stints," team chief Christian Horner explained, "(we) have decided that Tonio will continue as race driver at ... Germany."
A disappointed Klien, though - to the Austrian media - vowed to continue to 'do my best' in the Nurburgring third car.
Asked if he thought he had done a better job than Vitantonio Liuzzi in the first three grands prix, he continued: "It's not for me to judge.
"All I can say is that I gave my best efforts and I was happy with how I performed."