The Silkmen won 4-1 to set up a third-round clash with Cardiff but the victory was overshadowed by the alleged incident and a post-match fracas in the players' tunnel that was defused by police.
Town have reported the alleged incident to the police and the Football Association, and in a statement on their website, chief executive John Harris said: "We are very disappointed that this has overshadowed an excellent performance on the pitch, but we remain staunchly opposed to all forms of racism, which we believe has no place in football, or indeed anywhere."
The statement continued: "The club are aware that the matter has been reported to both the Police and the FA and will offer both it's fullest support in dealing with the matter."
A spokesman for Macclesfield told Press Association Sport that the Blue Square Bet Premier outfit hope to release a further statement on the incident on Monday.
The FA is already investigating an allegation that Town captain Nat Brown was racially abused by spectators in the original tie on December 19.
Manager Steve King said Brown had been called "a monkey" by some Barrow fans as he made his way out for the second half and referee David Webb included his version of events in his match report.
Source: PA
Source: PA