The alleged incident occurred during the Blues' 1-1 Champions League draw at Genk two weeks ago, the same game in which some of their travelling support aimed chants QPR defender Anton Ferdinand.
The Blues are investigating following a complaint from a Chelsea executive club member against a fellow supporter, who was sat in the Cristal Arena's exclusive 100-euro-a-seat section.
The unnamed fan - reportedly a well-dressed, middle-aged man - allegedly shouted 'They are bringing on the *****' when 22-year-old Sturridge came off the bench and is also alleged to have joined in the chanting about Ferdinand.
The club said in a statement: "We are investigating several complaints as part of our ongoing investigation into offensive behaviour at the game in Genk.
"We take all complaints of this nature extremely seriously and are working with supporters and police to identify those that took part.
"It can sometimes prove difficult to identify offenders at European away games as there are varying degrees of CCTV and TV footage available to us.
"However, we will continue to vigorously pursue any evidence of individuals or groups taking part in discriminatory behaviour and take the strongest action against them.
"Chelsea FC believes such activity shames the club and the game of football and we are fully committed to eradicating such offensive behaviour from the sport."