The Blues have rejected the Cherries' opening bid of £10m as they are looking for at least £12m for the Bosnian 'keeper.
Bournemouth are looking to sign a goalkeeper as current first choice 'keeper is Artur Boruc and at 36, Eddie Howe would like to add a younger face to his side.
The Cherries have also been leaking goals of late, conceding three goals in each of their last six games.
Begovic is currently second choice at Chelsea and has played 29 times for the Blues since joining from Stoke in 2015.