It could happen, Inter Milan management could fire Chivu at any time if he fails to meet expectations. I think Inter Milan has a chance to win the Scudetto, and that's still a realistic possibility at the moment. However, I doubt they'll perform as well in the Champions League as they did last season, given Chivu lack of experience. Meanwhile, we realize that Inter Milan management is not careless in appointing a manager, they see Chivu as worthy of holding this responsibility.
Inter Milan perhaps failed to convince better coaches to take over this head coach position after Inzaghi departure but it does not mean they will accept too poor performances from Chivu. The match with Juventus will be a big test for Chivu and Inter Milan under his management, and support from Inter Milan might drop considerably if Inter Milan loses this important direct match that has value like a 6-point match.
He still has a lot to prove himself to be, but he can do it. As I always say, to really see what a coach can give, you have to let him make purchases that he thinks can be good for the team and see if there are any improvements. But in this case, there was little to change, you just need to manage the players, who are of the highest level.
Inter Milan actually knew risk of believing in a young and inexperience coach like Chivu but I believe that they did not reduce their targets in the season like no Scudetto, no Champions League title. Chivu with 1 win, 1 defeat after two matches, has still been not in too bad situation in the club. However, it will change considerably if he loses in a coming big match with a direct competitor for Scudetto and top four, Juventus. This defeat if happens, will put Chivu under a lot of pressure, uncertainty in the club and perhaps a tough decision can be made, firing him after only three matches in Serie A, is not a kind of impossible scenario.