Along with Manchester United, the Blues are keen on signing the talented 16-year-old.
Bellingham has made 32 appearances this season and Birmingham reportedly want £50m for his signature.
But any move from Chelsea could have ramifications for their own academy player Tino Anjorin.
Goalsays Anjorin would re-consider signing a contract extension should Bellingham arrive at the West London club.
The 18-year-old has been stalling on penning a new five-year deal, and the acquisition of fellow attacking midfielder in Bellingham could sway him to look at other options.
Anjorin has 18 months remaining on his current contract.