Head coach Avram Grant wants the France international to bolster his attacking options with Didier Drogba on African Nations Cup duty later this month.
Anelka has taken Chelsea's eye after scoring 11 goals this season in a struggling side.
The London club made their first formal bid but it has been rejected by Bolton.
Former Arsenal striker Anelka signed a new four-year contract earlier this season but his superb form has seen him strongly linked with to a move to a bigger club.
Valued at about £10 million, only last week he revealed to a French newspaper that Bolton and Chelsea were in a dialogue about a possible move.
However manager Gary Megson played down claims a deal was imminent and following the FA Cup defeat by Sheffield United on Saturday he insisted there had been no formal approach for Anelka by any club.
A brief statement, posted on Chelsea's official website on Monday, read: "We can confirm we have made an official offer for the transfer of Nicolas Anelka, which is being discussed with Bolton Wanderers."