The Stoke stopper revealed he almost signed for City when Roberto Mancini was manager, suggesting that even the previous manager at the Etihad Stadium was looking to provide stronger competition for Joe Hart.
Manchester United and Manchester City were both heavily linked with the Bosnian international over the summer, and as recently as last month Stoke chairman Peter Coates said the club would consider offers.
But Begovic has now revealed City were the closest to winning the race for his signature.
He said: "As players, we are the last to find out things, probably our agents and club officials find out first without talking to us, then at the latter stage of negotiations they will tell us.
"As for Manchester City, when Mancini was on the bench, we were very close to an agreement.
"The transfer was not realised due to several reasons but the main one was the departure of the Italian manager."
Begovic refused to rule out a move to City in the future, but admitted it would be difficult to move just before the World Cup.
"Could I still move to City?" continued the Stoke keeper. "I really don't know if negotiations between clubs are still ongoing but for me probably the best thing would be to stay at Stoke City until the end of the season as in that way I would be truly ready for the World Cup in Brazil."
Begovic has been one of the most consistent keepers in the Premier League over the last few seasons, and City and United were both believed to be scouting him last season.
Atletico Madrid and Napoli were also linked with the 26-year-old stopper.