According to Il Messaggero, the Blues have officially offered around £43m for the 25-year-old.
The bid is described as ‘mind-boggling’ by the Italian outlet.
Antonio Conte is determined to land Koulibaly as he looks to bolster his defensive options in west London.
But Napoli are reluctant to sell, especially after already losing Gonzalo Higuain to Juventus this summer.
The Serie A outfit are trying their best to tie the Senegalese centre-half down to a new deal with improved terms and a release clause.
Koulibaly made 33 top flight appearances last term.
He has a further three years left to run on his current deal in Naples.