Why didn't Itachi tell Hiruzen (the 3rd Hokage) that Danzo attacked Shisui? So when we see Itachi and Shisui in the past, Shisui is the one who asks the 3rd hokage about trying to divert the Uchiha Coup on his own for which he get's approval by the third Hokage at which Danzo was also present. 

When Shisui tried to use genjuitsu on his clan, he's intercepted and attacked by Danzo, surely the hokage didn't approve of that. Shisui was able to tell Itachi of what happened, but why didn't Itachi tell the Hokage about Danzo's betrayal?
This is a speculation, but given that Danzo was the head of ANBU Black ops, it is most likely that they don't leave any trace whatsoever that can be tracked back to Danzo. Thus the only "proof" of Danzo attacking Shisui would be Shisui's own testament and Itachi's second hand testament. And given that Shishui's genjutsu, "koto amatsukami" is a complete hypnosis, who would believe Itachi?

Danzo can just claim that Itachi is under Shisui's genjutsu and that Shisui did that to blacksheep him. Danzo is a high official. Accussations towards him without any proof would result in heavy retaliation. Especially since by that time the Uchiha clan is already suspected of planning a coup.

This, I think, is why Itachi cannot tell Hiruzen. It would not solve anything. It will only smear Shisui's good name.