Following the game, which Macclesfield lost 3-2, several Cheltenham fans took to social media to speak out against the “disgusting” and “horrendous” alleged incidents.
Press Association Sport understands former England defender Campbell has told Macclesfield he will let the authorities deal with the matter.
Cheltenham were not available for immediate comment when contacted by Press Association Sport on Tuesday evening.
It is understood the FA has now opened an investigation after being made aware of the alleged abuse.
On Monday, Cheltenham had posted a statement on the club website “to remind all supporters visiting the Jonny-Rocks Stadium that it is against the law to shout or chant abuse on the grounds of ability or disability; age; gender; gender reassignment; marital status or civil partnership; pregnancy and maternity; race, nationality or ethnicity; religion or belief; or sexual orientation.”
The statement added: “Under the Equality Act 2010 it is an offence to make reference to the above protected characteristics in abusive terms and doing so could lead to arrest and prosecution.”
Cheltenham said the club were also “undertaking a thorough review of matchday operations in line with its Equality Policy”. Supporters have been urged to report any such abuse “to the nearest steward as quickly as possible.”
The statement concluded: “Please assist us in making the Jonny-Rocks Stadium a place where everyone can watch football without having to endure discriminatory abuse.”
Former Tottenham, Arsenal, Portsmouth and England defender Campbell was appointed as Macclesfield manager at the end of November 2018.
The 44-year-old had seen his side win their previous two league games before losing at Cheltenham.
The Silkmen are currently second from bottom of the table and are away at promotion-chasing Mansfield on Saturday.
Register for free to continue reading
Registration is a free and easy way to support our truly independent journalism
By registering, you will also enjoy limited access to Premium articles, exclusive newsletters, commenting, and virtual events with our leading journalists
Already have an account? sign in
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies