They are wary of losing Ivory Coast striker Didier Drogba, who has intimated he wants to leave and Anelka, 28, fits the bill as a suitable replacement.
Bolton will not be willing sellers but know the money they can recoup on the man they bought for £8m 14 months ago can help to rebuild a struggling squad.
Anelka has made no secret of his desire to play in the Champions League and would jump at the chance to move to Stamford Bridge.