The Blues supremo sees the 6ft 6in stopper as the perfect back up to number 1 Petr Cech.
Begović, 23, was widely expected to sign for the Blues last summer only to be told by Stoke the move was off as they were unwilling to sell at that time.
The 4 cap keeper has become more of a regular starter this season for the Potters, but it is believed he would still be interested in joining the Blues.
Chelsea were hoping a fee of around £5 million would be enough to secure the players signature but Tony Pulis is likely to ask for closer to £10 million.
It has been suggested the Blues would be willing to wait until the next transfer window if necessary to get their man.
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