Despite the departures of Andreas Christensen, Antonio Rudiger and Romelu Lukaku, head coach Thomas Tuchel is still waiting to make his first addition of the summer transfer window.
The Premier League giants have been linked with numerous centre-backs, including Sevilla's Jules Kounde and Juventus' Matthijs de Ligt.
However, it has been widely suggested that Manchester City's Nathan Ake is most likely to move to Stamford Bridge, returning for a second spell at the club.
In recent days, speculation has increased regarding the next move of Koulibaly with Chelsea known to want to bring another leadership figure into the group.
The Blues are allegedly prepared to meet the asking price of the Senegal international at a time when he has less than a year remaining on his contract.
According to French journalist Fabrice Hawkins, Chelsea are now closing in on wrapping up the transfer of the 31-year-old.
Hawkins says that Chelsea are ready to pay €40m (£33.91m) for Koulibaly, who is said to be ready to pen a three-year contract at the West Londoners.
The alleged development comes just days after Napoli director Cristian Giantuli claimed that Koulibaly had been provided with the opportunity to end his career in Naples.
Giantuli said: "Koulibaly is key player for us. There's no bid from any club. We've offered him new deal for €6m net salary per season plus a future role as part of the board.
"It's a huge proposal, €60m gross for Napoli - Koulibaly deserves it all. We didn't need to tell Koulibaly that we're not selling him to Juventus - because we consider Koulibaly untouchable and non-transferrable."
A further claim from journalist Nathan Gissing is that Tuchel would prefer to sign Koulibaly over Kounde, despite Chelsea's long-standing interest in the younger Frenchman.
Should Koulibaly make the decision to leave Napoli, he would be ending an eight-year association with the Serie A giants.
During that period, as many as 317 appearances have been made in all competitions, with the 2019-20 Coppa Italia the player's standout achievement when it comes to silverware.