Lukaku, on loan at Goodison Park from Chelsea, reportedly described the issue, which resulted in West Brom forward Anelka being charged by the Football Association on Tuesday, as "not such a big deal" in a video briefly posted on Everton's official website.
The video was later removed and Everton have clarified that they do not share the striker's views, with director of communications Alan Myers telling the BBC that was "not the opinion held by the football club".
Anelka was embroiled in controversy following his decision to make the 'quenelle' salute, which is said to carry anti-Semitic connotations in his native France, after scoring in West Brom's 3-3 draw at West Ham last month.
He defended the gesture by saying it was merely a dedication to his friend and comedian Dieudonne M'bala M'bala, but the incident drew widespread criticism.
Lukaku reportedly said: "I don't think he [Anelka] should be banned; he was just supporting a comedian in France. We don't have to make such a big deal about it."
Myers added: "We moved quickly to remove any interview that may have been offensive to anyone.
"The last thing we want to do is cause offence to anyone."
Anelka is facing a minimum ban of five matches following the FA's decision to charge him.