Whitby Town lost 2-1 during Saturday's game against Scarborough Athletic during the Yorkshire coast derby in the Northern Premier League, Premier Division
The match was marred by racist language from the stands.
Whitby Town confirmed a full investigation is taking place and "appropriate action" will be taken accordingly.
Issuing a statement on Twitter, the club said: "Following yesterday's game with Scarborough Athletic, Whitby Town would like to issue the following statement:
"The Board are aware of an alleged incident during the game with Scarborough Athletic at the Flamingo Land Stadium on Saturday, November 30, which involved racist language being used.
"We are investigating the full details and we will take appropriate action accordingly.
"Racism and other discriminative behaviour has no place at Whitby Town Football Club and will not be tolerated under any circumstances."