The allegation from Macclesfield followed the FA Cup second-round match between the two clubs at Holker Street which ended 1-1.
Macclesfield manager Steve King complained that Brown had been called "a monkey" by some Barrow fans as he made his way out for the second half.
Referee David Webb included his version of events in his report of the match, which will go to a December 29 replay to determine who will face npower Championship side Cardiff in the third round.
A spokesman for the FA said: "We are aware of the incident and it was referenced in the referee's report.
"We are now liaising with both clubs and the police."
Barrow chairman Brian Keen said the allegations of racism would be dealt with seriously by the Blue Square Bet Premier side.
"It's just unacceptable and it will be looked into," Keen told BBC Radio Cumbria.
"The results will be known to people once we find out who or what it was. We at Barrow won't have it and we will look into it really seriously."
Speaking after the match, King claimed that defender Brown was the subject of racial abuse by home fans.
King said: "He got called a monkey by some of the fans behind the goal. He said to the ref 'he's called me a monkey' and that's not acceptable.
"I heard it when I was coming out with Nat. It seems to be happening everywhere."
The alleged incident is the latest allegation of abuse in football following several other high-profile incidents.
Several England Under-21 players were targeted with racist chanting by Serbia fans during their Euro 2013 qualifier in October, while former England captain John Terry was banned for four matches by the FA, before being cleared of all criminal charges, following an exchange of words with QPR defender Anton Ferdinand.
Liverpool striker Luis Suarez was given an eight-game ban and a £40,000 fine after he was found guilty of racially abusing Manchester United's Patrice Evra last season.
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