The Argentine goalkeeper, 32, completed a move to the Etihad Stadium from Malaga earlier this month.
Caballero is expected to play second fiddle to England international Hart, but aims to convince Manuel Pellegrini - his former coach at Malaga - he is worthy of a regular starting berth.
"I know the situation now but I will fight to play. The decision will be down to the coach, the boss. I have trained well and I will fight," said Caballero.
"I know my position, but I want to try hard to fight for this position. Joe is a fantastic goalkeeper, but I've come to play - I want to try and take my opportunity.
"It will be difficult, yes, but City are a big club and it's a big opportunity. I want to try in every way I can to play."
Caballero acknowledged that Pellegrini was a major factor in his decision to join City.
"It was a crucial part of me coming to City knowing him from before and having played for him. He was a very important figure in me coming to Manchester City," added the experienced stopper.
"I had no doubt about the move. I think it was necessary for my career, for my life and for my family. It's fantastic for me.
“He hasn't given me any promises. No, I only know that he wanted me. I came here for the best opportunity for my career.
“I don't know if I play in the cup or other tournaments, I don't know anything. But of course I am confident.
"Goalkeepers are a different type of player and I believe in myself."