The Blues boss saw his first offer for the Bosnian international rejected, but is now hoping to be able to up the ante on negotiations in a bid to bring in a swift replacement for Petr Cech.
Stoke had initially been hoping to receive an offer around the £8m mark, while reports suggest that they would also be open to taking a player on loan from Chelsea, too.
But The Telegraph believe Mourinho will come back with an improved offer in the next 48 hours.
Veteran stopper Petr Cech finally completed his £11m move to the Emirates on Monday afternoon, and not wanting to hang around, it is thought that Mourinho will offer to double the wage Begovic earns at the Britannia.
However, convincing him to play second fiddle to Thibaut Courtois.