The Bosnia international's form for the Potters has seen him linked with a number of clubs but it appears Mancini was prepared to take that interest a step further before he was sacked at the end of last season.
"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," Begovic is quoted as saying by Bosnian newspaper 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."
Begovic, who on Saturday became only the fifth goalkeeper to score in the Premier League with a goal after just 13 seconds in the 1-1 draw with Southampton, is now focusing on his country's first appearance at a World Cup next summer.
"Could I still move to City? I really don't know if negotiations between clubs are still ongoing," he added.
"But for me probably the best thing would be to stay at Stoke until the end of the season as in that way I would be truly ready for World Cup in Brazil."
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