The Stoke City goalkeeper is widely regarded as one of the best stoppers in the Premier League, with several leading clubs said to be interested in his services.
During the last transfer window, David Platt - who worked as Mancini's assistant manager at Manchester City prior to the Italian being dismissed in May - told Al Jazeera that the former Inter boss wanted to replace Joe Hart with Begovic as his number one keeper.
And Bosnia-Herzegovina international Begovic has now confirmed that he was on the verge of a switch to the Etihad Stadium, only for the deal to be scuppered by Mancini's exit.
"As for Manchester City, when Mancini was on the bench, we were very close to an agreement," the 26-year-old told Bosnian newspaper Dnevni Avaz.
"The transfer was not realised due to several reasons but the main one was the departure of the Italian manager."
Manchester City's goalkeeping position has been a hot topic in recent weeks, with new boss Manuel Pellegrini deciding to drop Hart in favour of Romanian back-up Costel Pantilimon for Saturday's game against Norwich City after a number of high-profile errors by the England international.
This has led to speculation that Pellegrini could sign a new keeper in the next transfer window, although Begovic does not expect to move to Manchester in January as he gears up for his country's maiden FIFA World Cup campaign.
"Could I still move to City? 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 World Cup in Brazil," he added.
Begovic went on to praise Stoke manager Mark Hughes, who took over from Tony Pulis in the close-season.
"He is wonderful, a real manager who takes care of everything … I think Stoke did a great job of bringing Hughes and we are playing much better football."
