The Cherries are said to be interested in the Bosnian international, who has only made 4 appearances for Chelsea this season.
Moving to Stamford Bridge in 2015, the experienced shot stopper signed from Stoke City for £8 million.
The 29-year-old has found opportunities hard to come by at the West London club, with all of his appearances this season coming in cup games.
He is likely to make an appearance this weekend, however, as Chelsea face up against Brentford in the FA Cup, as the Blues host their West London neighbours.
Begovic made seven appearances for The Cherries back in 2007 while on loan from Portsmouth. He has also played for Ipswich Town.
It is unclear at this moment whether Bournemouth boss Eddie Howe would be looking to sign Begovic as a rival for current first choice Artur Boruc, or as a back up.
It is unlikely, however, that Begovic will want to move from one team’s substitute bench to another, without some guarantees about first team opportunities.