Caballero was preferred over the England stopper for the Community Shield clash with Arsenal at Wembley on Sunday, but the former Malaga man failed to impress.
The Gunners ran out 3-0 winners as goals from Santi Cazorla, Aaron Ramsey and Olivier Giroud gave Arsene Wenger's side a convincing win against their title rivals.
Caballero has replaced Costel Pantilimon in the City squad, with the tall Romanian joining Sunderland on a free transfer, but the Argentinian keeper thinks that he may has blown his audition to cement to his place as City's first choice at Wembley.
Joe Hart suffered something of a dip in form last season and he was dropped in favour of Pantilimon, but the arrival of Caballero was completed with the intention of giving the Three Lions custodian more serious competition for his place in the City lineup.
However, Caballero accepted that he didn't cover himself in glory against the Gunners on Sunday afternoon.
Speaking to the Manchester Evening News, he said: “I have been training well but there is no doubt that this was not one of my best games.
“They attacked three or four times and they scored three goals. But the decision is for the manager. Defeat hurts - it is a final and I think when it was 2-0 we could have got back into it.
“My ambition is to fight for the position and to be there when the manager needs me. I feel good. It was an important match for me. I have a chance to play and I am happy for this, not for the score. I will train for the next opportunity.
“I want to be number one. But this is today. I will train for the next decision to see who starts next week. Both of us will train for that chance. But I came to fight for the number one spot.”
Caballero is one of four arrivals at the Etihad this summer, as Eliaquim Mangala signed for the club for £32 million on Monday to join the goalkeeper alongside Fernando and Bacary Sagna as the City new-boys.
Meanwhile Manuel Pellegrini is set to have a whole host of first-team stars return to training this week in what will be a huge boost the the squad ahead of the new campaign.
Pablo Zabaleta, Bacary Sagna, Vincent Kompany, Martin Demichelis, Fernandinho, Frank Lampard and Sergio Aguero will all train this week, although the latter could miss the opener against Newcastle United as he eases back to full fitness.