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."