The Daily Mirror reports that City are ready to make a bid of up to £15million for the Bosnia & Herzegovina international.
England international Joe Hart could also move to Stoke on loan for the second half of the season as part of a deal.
Begovic was also strongly linked with a move to City while Roberto Mancini was in charge at Eastlands.
The 26-year-old has over two years of his contract left to run and Stoke are expected to hold out for a massive fee for the goalkeeper.
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Summer 2016 is set to be a very exciting time for football fans, as this is when the long awaited Euro 2016 tournament kicks off.
Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.