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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The start of the 2017-18 Premier League season is just around the corner and is shaping up to be one of the most competitive yet.
As always it is the so called “big clubs” that have splashed the most cash in the transfer window so far this summer. Manchester United or City? Arsenal or Tottenham? Who are the winners and losers of the transfer window so far?