Mancini's former assistant David Platt stated earlier this summer that they wanted Begovic, and now the Bosnian has also revealed a deal was close.
"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," he told Dnevni Avaz.
"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."
He added: "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."
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